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| Boilerpipe Text | Central Limit Theorem
Type
Theorem
Field
Probability theory
Statement
The scaled sum of a sequence of
i.i.d. random variables
with finite positive
variance
converges in distribution to the
normal distribution
.
Generalizations
Lindeberg's CLT
In
probability theory
, the
central limit theorem
(
CLT
) states that, under appropriate conditions, the
distribution
of a normalized version of the sample mean converges to a
standard normal distribution
. This holds even if the original variables themselves are not
normally distributed
. There are several versions of the CLT, each applying in the context of different conditions.
The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated in the 1920s.
[
1
]
In
statistics
, the CLT can be stated as: let
denote a
statistical sample
of size
from a population with
expected value
(average)
and finite positive
variance
, and let
denote the sample mean (which is itself a
random variable
). Then the
limit as
of the distribution
of
is a normal distribution with mean
and variance
.
[
2
]
In other words, suppose that a large sample of
observations
is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (
arithmetic mean
) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the
probability distribution
of these averages will closely approximate a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be
independent and identically distributed
(i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the
binomial distribution
, is the
de MoivreâLaplace theorem
.
Independent sequences
[
edit
]
Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.
[
3
]
Let
be a sequence of
i.i.d. random variables
having a distribution with
expected value
given by
and finite
variance
given by
Suppose we are interested in the
sample average
By the
law of large numbers
, the sample average
converges almost surely
(and therefore also
converges in probability
) to the expected value
as
The classical central limit theorem describes the size and the distributional form of the
stochastic
fluctuations around the deterministic number
during this convergence. More precisely, it states that as
gets larger, the distribution of the normalized mean
, i.e. the difference between the sample average
and its limit
scaled by the factor
, approaches the
normal distribution
with mean
and variance
For large enough
the distribution of
gets arbitrarily close to the normal distribution with mean
and variance
The usefulness of the theorem is that the distribution of
approaches normality regardless of the shape of the distribution of the individual
Formally, the theorem can be stated as follows:
In the case
convergence in distribution means that the
cumulative distribution functions
of
converge pointwise to the cdf of the
distribution: for every real number
where
is the standard normal cdf evaluated at
The convergence is uniform in
in the sense that
where
denotes the
supremum
(i.e. least upper bound) of the set.
[
5
]
In this variant of the central limit theorem the random variables
have to be independent, but not necessarily identically distributed. The theorem also requires that random variables
have
moments
of some order
,
and that the rate of growth of these moments is limited by the Lyapunov condition given below.
Lyapunov CLT
[
6
]
â
Suppose
is a sequence of independent random variables, each with finite expected value
and variance
.
Define
If for some
,
Lyapunovâs condition
is satisfied, then a sum of
converges in distribution to a standard normal random variable, as
goes to infinity:
In practice it is usually easiest to check Lyapunov's condition for
.
If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
Lindeberg (-Feller) CLT
[
edit
]
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from
Lindeberg
in 1920).
Suppose that for every
,
where
is the
indicator function
. Then the distribution of the standardized sums
converges towards the standard normal distribution
.
CLT for the sum of a random number of random variables
[
edit
]
Rather than summing an integer number
of random variables and taking
, the sum can be of a random number
of random variables, with conditions on
. For example, the following theorem is Corollary 4 of Robbins (1948). It assumes that
is asymptotically normal (Robbins also developed other conditions that lead to the same result).
Multidimensional CLT
[
edit
]
Proofs that use characteristic functions can be extended to cases where each individual
is a
random vector
in
,
with mean vector
and
covariance matrix
(among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a
multivariate normal distribution
.
[
9
]
Summation of these vectors is done component-wise.
For
let
be independent random vectors. The sum of the random vectors
is
and their average is
Therefore,
The multivariate central limit theorem states that
where the
covariance matrix
is equal to
The multivariate central limit theorem can be proved using the
CramĂ©râWold theorem
.
[
9
]
The rate of convergence is given by the following
BerryâEsseen
type result:
It is unknown whether the factor
is necessary.
[
11
]
The generalized central limit theorem
[
edit
]
The generalized central limit theorem (GCLT) was an effort of multiple mathematicians (
Sergei Bernstein
,
Jarl Waldemar Lindeberg
,
Paul LĂ©vy
,
William Feller
,
Andrey Kolmogorov
, and others) over the period from 1920 to 1937.
[
12
]
The first published complete proof of the GCLT was in 1937 by Paul LĂ©vy in French.
[
13
]
An English language version of the complete proof of the GCLT is available in the translation of
Boris Vladimirovich Gnedenko
and Kolmogorov's 1954 book.
[
14
]
The statement of the GCLT is as follows:
[
15
]
Statement of
GCLT
â
A non-degenerate random variable
Z
is
α
-stable
for some
0 <
α
†2
if and only if there is an independent, identically distributed sequence of random variables
X
1
,
X
2
,
X
3
, ..., and constants
a
n
> 0
,
b
n
â â
with
Here,
'
â
'
means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy
F
n
(
y
) â
F
(
y
)
at all continuity points of
F
.
In other words, if sums of independent, identically distributed random variables converge in distribution to some
Z
, then
Z
must be a
stable distribution
.
Dependent processes
[
edit
]
CLT under weak dependence
[
edit
]
A useful generalization of a sequence of independent, identically distributed random variables is a
mixing
random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially
strong mixing
(also called α-mixing) defined by
where
is so-called
strong mixing coefficient
.
A simplified formulation of the central limit theorem under strong mixing is:
[
16
]
In fact,
where the series converges absolutely.
The assumption
cannot be omitted, since the asymptotic normality fails for
where
are another
stationary sequence
.
There is a stronger version of the theorem:
[
17
]
the assumption
is replaced with
,
and the assumption
is replaced with
Existence of such
ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (
Bradley 2007
).
Martingale difference CLT
[
edit
]
Theorem
â
Let a
martingale
satisfy
then
converges in distribution to
as
.
[
18
]
[
19
]
Proof of classical CLT
[
edit
]
The central limit theorem has a proof using
characteristic functions
.
[
20
]
It is similar to the proof of the (weak)
law of large numbers
.
Assume
are independent and identically distributed random variables, each with mean
and finite variance
.
The sum
has
mean
and
variance
.
Consider the random variable
where in the last step we defined the new random variables
,
each with zero mean and unit variance
(
).
The
characteristic function
of
is given by
where in the last step we used the fact that all of the
are identically distributed. The characteristic function of
is, by
Taylor's theorem
,
where
is "
little
o
notation
" for some function of
that goes to zero more rapidly than
.
By the limit of the
exponential function
(
),
the characteristic function of
equals
All of the higher order terms vanish in the limit
.
The right hand side equals the characteristic function of a standard normal distribution
, which implies through
LĂ©vy's continuity theorem
that the distribution of
will approach
as
.
Therefore, the
sample average
is such that
converges to the normal distribution
,
from which the central limit theorem follows.
Convergence to the limit
[
edit
]
The central limit theorem gives only an
asymptotic distribution
. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
[
citation needed
]
The convergence in the central limit theorem is
uniform
because the limiting cumulative distribution function is continuous. If the third central
moment
exists and is finite, then the speed of convergence is at least on the order of
(see
BerryâEsseen theorem
).
Stein's method
[
21
]
can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.
[
22
]
The convergence to the normal distribution is monotonic, in the sense that the
entropy
of
increases
monotonically
to that of the normal distribution.
[
23
]
The central limit theorem applies in particular to sums of independent and identically distributed
discrete random variables
. A sum of
discrete random variables
is still a
discrete random variable
, so that we are confronted with a sequence of
discrete random variables
whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the
normal distribution
). This means that if we build a
histogram
of the realizations of the sum of
n
independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as
n
approaches infinity; this relation is known as
de MoivreâLaplace theorem
. The
binomial distribution
article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
Common misconceptions
[
edit
]
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.
[
24
]
[
25
]
[
26
]
These include:
The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of
iid
random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a
sampling distribution
formed from different values of means (or sums) of such random variables.
The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the
GlivenkoâCantelli theorem
.
The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,
[
27
]
allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See
Z-test
for where the approximation holds.
Relation to the law of large numbers
[
edit
]
The
law of large numbers
as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of
S
n
as
n
approaches infinity?" In mathematical analysis,
asymptotic series
are one of the most popular tools employed to approach such questions.
Suppose we have an asymptotic expansion of
:
Dividing both parts by
Ï
1
(
n
)
and taking the limit will produce
a
1
, the coefficient of the highest-order term in the expansion, which represents the rate at which
f
(
n
)
changes in its leading term.
Informally, one can say: "
f
(
n
)
grows approximately as
a
1
Ï
1
(
n
)
". Taking the difference between
f
(
n
)
and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about
f
(
n
)
:
Here one can say that the difference between the function and its approximation grows approximately as
a
2
Ï
2
(
n
)
. The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.
Informally, something along these lines happens when the sum,
S
n
, of independent identically distributed random variables,
X
1
, ...,
X
n
, is studied in classical probability theory.
[
citation needed
]
If each
X
i
has finite mean
Ό
, then by the law of large numbers,
â
S
n
/
n
â
â
Ό
.
[
28
]
If in addition each
X
i
has finite variance
Ï
2
, then by the central limit theorem,
where
Ο
is distributed as
N
(0,
Ï
2
)
. This provides values of the first two constants in the informal expansion
In the case where the
X
i
do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
or informally
Distributions
Î
which can arise in this way are called
stable
.
[
29
]
Clearly, the normal distribution is stable, but there are also other stable distributions, such as the
Cauchy distribution
, for which the mean or variance are not defined. The scaling factor
b
n
may be proportional to
n
c
, for any
c
â„
â
1
/
2
â
; it may also be multiplied by a
slowly varying function
of
n
.
[
30
]
[
31
]
The
law of the iterated logarithm
specifies what is happening "in between" the
law of large numbers
and the central limit theorem. Specifically it says that the normalizing function
â
n
log log
n
, intermediate in size between
n
of the law of large numbers and
â
n
of the central limit theorem, provides a non-trivial limiting behavior.
Alternative statements of the theorem
[
edit
]
The
density
of the sum of two or more independent variables is the
convolution
of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov
[
32
]
for a particular local limit theorem for sums of
independent and identically distributed random variables
.
Characteristic functions
[
edit
]
Since the
characteristic function
of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
An equivalent statement can be made about
Fourier transforms
, since the characteristic function is essentially a Fourier transform.
Calculating the variance
[
edit
]
Let
S
n
be the sum of
n
random variables. Many central limit theorems provide conditions such that
S
n
/
â
Var(
S
n
)
converges in distribution to
N
(0,1)
(the normal distribution with mean 0, variance 1) as
n
â â
. In some cases, it is possible to find a constant
Ï
2
and function
f(n)
such that
S
n
/(Ï
â
nâ
f
(
n
)
)
converges in distribution to
N
(0,1)
as
n
â â
.
Products of positive random variables
[
edit
]
The
logarithm
of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a
log-normal distribution
. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different
random
factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called
Gibrat's law
.
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.
[
34
]
Beyond the classical framework
[
edit
]
Asymptotic normality, that is,
convergence
to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
Theorem
â
There exists a sequence
Δ
n
â 0
for which the following holds. Let
n
â„ 1
, and let random variables
X
1
, ...,
X
n
have a
log-concave
joint density
f
such that
f
(
x
1
, ...,
x
n
) =
f
(|
x
1
|, ..., |
x
n
|)
for all
x
1
, ...,
x
n
, and
E(
X
2
k
) = 1
for all
k
= 1, ...,
n
. Then the distribution of
is
Δ
n
-close to
in the
total variation distance
.
[
35
]
These two
Δ
n
-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Another example:
f
(
x
1
, ...,
x
n
) = const · exp(â(|
x
1
|
α
+ ⯠+ |
x
n
|
α
)
ÎČ
)
where
α
> 1
and
αÎČ
> 1
. If
ÎČ
= 1
then
f
(
x
1
, ...,
x
n
)
factorizes into
const · exp (â|
x
1
|
α
) ⊠exp(â|
x
n
|
α
),
which means
X
1
, ...,
X
n
are independent. In general, however, they are dependent.
The condition
f
(
x
1
, ...,
x
n
) =
f
(|
x
1
|, ..., |
x
n
|)
ensures that
X
1
, ...,
X
n
are of zero mean and
uncorrelated
;
[
citation needed
]
still, they need not be independent, nor even
pairwise independent
.
[
citation needed
]
By the way, pairwise independence cannot replace independence in the classical central limit theorem.
[
36
]
Here is a
BerryâEsseen
type result.
Theorem
â
Let
X
1
, ...,
X
n
satisfy the assumptions of the previous theorem, then
[
37
]
for all
a
<
b
; here
C
is a
universal (absolute) constant
. Moreover, for every
c
1
, ...,
c
n
â
R
such that
c
2
1
+ ⯠+
c
2
n
= 1
,
The distribution of
â
X
1
+ ⯠+
X
n
/
â
n
â
need not be approximately normal (in fact, it can be uniform).
[
38
]
However, the distribution of
c
1
X
1
+ ⯠+
c
n
X
n
is close to
(in the total variation distance) for most vectors
(
c
1
, ...,
c
n
)
according to the uniform distribution on the sphere
c
2
1
+ ⯠+
c
2
n
= 1
.
Lacunary trigonometric series
[
edit
]
Theorem (
Salem
â
Zygmund
)
â
Let
U
be a random variable distributed uniformly on
(0,2Ï)
, and
X
k
=
r
k
cos(
n
k
U
+
a
k
)
, where
n
k
satisfy the lacunarity condition: there exists
q
> 1
such that
n
k
+ 1
â„
qn
k
for all
k
,
r
k
are such that
0 â€
a
k
< 2Ï
.
Then
[
39
]
[
40
]
converges in distribution to
.
Theorem
â
Let
A
1
, ...,
A
n
be independent random points on the plane
R
2
each having the two-dimensional standard normal distribution. Let
K
n
be the
convex hull
of these points, and
X
n
the area of
K
n
Then
[
41
]
converges in distribution to
as
n
tends to infinity.
The same also holds in all dimensions greater than 2.
The
polytope
K
n
is called a Gaussian
random polytope
.
A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.
[
42
]
Linear functions of orthogonal matrices
[
edit
]
A linear function of a matrix
M
is a linear combination of its elements (with given coefficients),
M
⊠tr(
AM
)
where
A
is the matrix of the coefficients; see
Trace (linear algebra)#Inner product
.
A random
orthogonal matrix
is said to be distributed uniformly, if its distribution is the normalized
Haar measure
on the
orthogonal group
O(
n
,
R
)
; see
Rotation matrix#Uniform random rotation matrices
.
Theorem
â
Let
M
be a random orthogonal
n
Ă
n
matrix distributed uniformly, and
A
a fixed
n
Ă
n
matrix such that
tr(
AA
*) =
n
, and let
X
= tr(
AM
)
. Then
[
43
]
the distribution of
X
is close to
in the total variation metric up to
[
clarification needed
]
â
2
â
3
/
n
â 1
â
.
Theorem
â
Let random variables
X
1
,
X
2
, ... â
L
2
(Ω)
be such that
X
n
â 0
weakly
in
L
2
(Ω)
and
X
n
â 1
weakly in
L
1
(Ω)
. Then there exist integers
n
1
<
n
2
< âŻ
such that
converges in distribution to
as
k
tends to infinity.
[
44
]
Random walk on a crystal lattice
[
edit
]
The central limit theorem may be established for the simple
random walk
on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.
[
45
]
[
46
]
Applications and examples
[
edit
]
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample
statistics
to the normal distribution in controlled experiments.
Comparison of probability density functions
p
(
k
)
for the sum of
n
fair 6-sided dice to show their convergence to a normal distribution with increasing
n
, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the
chi-squared
values that quantify the goodness of the fit (the fit is good if the reduced
chi-squared
value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/
â
n
), which is called the standard deviation of the mean (since it refers to the spread of sample means).
Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.
Regression analysis
, and in particular
ordinary least squares
, specifies that a
dependent variable
depends according to some function upon one or more
independent variables
, with an additive
error term
. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.
Other illustrations
[
edit
]
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.
[
47
]
Dutch mathematician
Henk Tijms
writes:
[
48
]
The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician
Abraham de Moivre
who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician
Pierre-Simon Laplace
rescued it from obscurity in his monumental work
Théorie analytique des probabilités
, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician
Aleksandr Lyapunov
defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.
Sir
Francis Galton
described the Central Limit Theorem in this way:
[
49
]
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by
George PĂłlya
in 1920 in the title of a paper.
[
50
]
[
51
]
PĂłlya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word
central
in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".
[
51
]
The abstract of the paper
On the central limit theorem of calculus of probability and the problem of moments
by PĂłlya
[
50
]
in 1920 translates as follows.
The occurrence of the Gaussian probability density
1 =
e
â
x
2
in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by
Liapounoff
. ...
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as
Cauchy
's,
Bessel
's and
Poisson
's contributions, is provided by Hald.
[
52
]
Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by
von Mises
,
PĂłlya
,
Lindeberg
,
LĂ©vy
, and
Cramér
during the 1920s, are given by Hans Fischer.
[
53
]
Le Cam describes a period around 1935.
[
51
]
Bernstein
[
54
]
presents a historical discussion focusing on the work of
Pafnuty Chebyshev
and his students
Andrey Markov
and
Aleksandr Lyapunov
that led to the first proofs of the CLT in a general setting.
A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of
Alan Turing
's 1934 Fellowship Dissertation for
King's College
at the
University of Cambridge
. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.
[
55
]
Asymptotic equipartition property
Asymptotic distribution
Bates distribution
Benford's law
â result of extension of CLT to product of random variables.
BerryâEsseen theorem
Central limit theorem for directional statistics
â Central limit theorem applied to the case of directional statistics
Delta method
â to compute the limit distribution of a function of a random variable.
ErdĆsâKac theorem
â connects the number of prime factors of an integer with the normal probability distribution
FisherâTippettâGnedenko theorem
â limit theorem for extremum values (such as
max{
X
n
}
)
IrwinâHall distribution
Markov chain central limit theorem
Normal distribution
Tweedie convergence theorem
â a theorem that can be considered to bridge between the central limit theorem and the
Poisson convergence theorem
[
56
]
Donsker's theorem
^
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by Carl McTague |
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## Contents
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- [(Top)](https://en.wikipedia.org/wiki/Central_limit_theorem)
- [1 Independent sequences](https://en.wikipedia.org/wiki/Central_limit_theorem#Independent_sequences)
Toggle Independent sequences subsection
- [1\.1 Classical CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Classical_CLT)
- [1\.2 Lyapunov CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT)
- [1\.3 Lindeberg (-Feller) CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Lindeberg_\(-Feller\)_CLT)
- [1\.4 CLT for the sum of a random number of random variables](https://en.wikipedia.org/wiki/Central_limit_theorem#CLT_for_the_sum_of_a_random_number_of_random_variables)
- [1\.5 Multidimensional CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Multidimensional_CLT)
- [2 The generalized central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem#The_generalized_central_limit_theorem)
- [3 Dependent processes](https://en.wikipedia.org/wiki/Central_limit_theorem#Dependent_processes)
Toggle Dependent processes subsection
- [3\.1 CLT under weak dependence](https://en.wikipedia.org/wiki/Central_limit_theorem#CLT_under_weak_dependence)
- [3\.2 Martingale difference CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Martingale_difference_CLT)
- [4 Remarks](https://en.wikipedia.org/wiki/Central_limit_theorem#Remarks)
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- [4\.1 Proof of classical CLT](https://en.wikipedia.org/wiki/Central_limit_theorem#Proof_of_classical_CLT)
- [4\.2 Convergence to the limit](https://en.wikipedia.org/wiki/Central_limit_theorem#Convergence_to_the_limit)
- [4\.3 Common misconceptions](https://en.wikipedia.org/wiki/Central_limit_theorem#Common_misconceptions)
- [4\.4 Relation to the law of large numbers](https://en.wikipedia.org/wiki/Central_limit_theorem#Relation_to_the_law_of_large_numbers)
- [4\.5 Alternative statements of the theorem](https://en.wikipedia.org/wiki/Central_limit_theorem#Alternative_statements_of_the_theorem)
- [4\.5.1 Density functions](https://en.wikipedia.org/wiki/Central_limit_theorem#Density_functions)
- [4\.5.2 Characteristic functions](https://en.wikipedia.org/wiki/Central_limit_theorem#Characteristic_functions)
- [4\.6 Calculating the variance](https://en.wikipedia.org/wiki/Central_limit_theorem#Calculating_the_variance)
- [5 Extensions](https://en.wikipedia.org/wiki/Central_limit_theorem#Extensions)
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- [5\.1 Products of positive random variables](https://en.wikipedia.org/wiki/Central_limit_theorem#Products_of_positive_random_variables)
- [6 Beyond the classical framework](https://en.wikipedia.org/wiki/Central_limit_theorem#Beyond_the_classical_framework)
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- [6\.1 Convex body](https://en.wikipedia.org/wiki/Central_limit_theorem#Convex_body)
- [6\.2 Lacunary trigonometric series](https://en.wikipedia.org/wiki/Central_limit_theorem#Lacunary_trigonometric_series)
- [6\.3 Gaussian polytopes](https://en.wikipedia.org/wiki/Central_limit_theorem#Gaussian_polytopes)
- [6\.4 Linear functions of orthogonal matrices](https://en.wikipedia.org/wiki/Central_limit_theorem#Linear_functions_of_orthogonal_matrices)
- [6\.5 Subsequences](https://en.wikipedia.org/wiki/Central_limit_theorem#Subsequences)
- [6\.6 Random walk on a crystal lattice](https://en.wikipedia.org/wiki/Central_limit_theorem#Random_walk_on_a_crystal_lattice)
- [7 Applications and examples](https://en.wikipedia.org/wiki/Central_limit_theorem#Applications_and_examples)
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- [7\.1 Regression](https://en.wikipedia.org/wiki/Central_limit_theorem#Regression)
- [7\.2 Other illustrations](https://en.wikipedia.org/wiki/Central_limit_theorem#Other_illustrations)
- [8 History](https://en.wikipedia.org/wiki/Central_limit_theorem#History)
- [9 See also](https://en.wikipedia.org/wiki/Central_limit_theorem#See_also)
- [10 Notes](https://en.wikipedia.org/wiki/Central_limit_theorem#Notes)
- [11 References](https://en.wikipedia.org/wiki/Central_limit_theorem#References)
- [12 External links](https://en.wikipedia.org/wiki/Central_limit_theorem#External_links)
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# Central limit theorem
42 languages
- [Afrikaans](https://af.wikipedia.org/wiki/Sentrale_limietstelling "Sentrale limietstelling â Afrikaans")
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ۚ۱ÙÙŰ© ۧÙÙÙۧÙŰ© ۧÙÙ
۱ÙŰČÙŰ© â Arabic")
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â Greek")
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۱کŰČÛ â Persian")
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- [Français](https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_central_limite "ThĂ©orĂšme central limite â French")
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- [Magyar](https://hu.wikipedia.org/wiki/Centr%C3%A1lis_hat%C3%A1reloszl%C3%A1s-t%C3%A9tel "CentrĂĄlis hatĂĄreloszlĂĄs-tĂ©tel â Hungarian")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Teorema_limit_pusat "Teorema limit pusat â Indonesian")
- [Ăslenska](https://is.wikipedia.org/wiki/H%C3%B6fu%C3%B0setning_t%C3%B6lfr%C3%A6%C3%B0innar "HöfuĂ°setning tölfrĂŠĂ°innar â Icelandic")
- [Italiano](https://it.wikipedia.org/wiki/Teoremi_del_limite_centrale "Teoremi del limite centrale â Italian")
- [æ„æŹèȘ](https://ja.wikipedia.org/wiki/%E4%B8%AD%E5%BF%83%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86 "äžćżæ„”éćźç â Japanese")
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- [Nederlands](https://nl.wikipedia.org/wiki/Centrale_limietstelling "Centrale limietstelling â Dutch")
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- [Simple English](https://simple.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem â Simple English")
- [Shqip](https://sq.wikipedia.org/wiki/Teorema_Q%C3%ABndrore_Limite "Teorema QĂ«ndrore Limite â Albanian")
- [ĐĄŃĐżŃĐșĐž / srpski](https://sr.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0 "ĐŠĐ”ĐœŃŃĐ°Đ»ĐœĐ° ĐłŃĐ°ĐœĐžŃĐœĐ° ŃĐ”ĐŸŃĐ”ĐŒĐ° â Serbian")
- [Sunda](https://su.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem â Sundanese")
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- [TĂŒrkçe](https://tr.wikipedia.org/wiki/Merkez%C3%AE_limit_teoremi "MerkezĂź limit teoremi â Turkish")
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۳ۊÙÛ Ű§Ű«ŰšŰ§ŰȘÛ â Urdu")
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From Wikipedia, the free encyclopedia
Fundamental theorem in probability theory and statistics
| | |
|---|---|
| [](https://en.wikipedia.org/wiki/File:IllustrationCentralTheorem.png) | |
| Type | [Theorem](https://en.wikipedia.org/wiki/Theorem "Theorem") |
| Field | [Probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") |
| Statement | The scaled sum of a sequence of [i.i.d. random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables") with finite positive [variance](https://en.wikipedia.org/wiki/Variance "Variance") converges in distribution to the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"). |
| Generalizations | [Lindeberg's CLT](https://en.wikipedia.org/wiki/Lindeberg%27s_condition "Lindeberg's condition") |
In [probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory"), the **central limit theorem** (**CLT**) states that, under appropriate conditions, the [distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") of a normalized version of the sample mean converges to a [standard normal distribution](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution "Normal distribution"). This holds even if the original variables themselves are not [normally distributed](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"). There are several versions of the CLT, each applying in the context of different conditions.
The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated in the 1920s.[\[1\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">[<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears. \(July_2023\)">page needed</span>]]</i>]</sup>-1)
In [statistics](https://en.wikipedia.org/wiki/Statistics "Statistics"), the CLT can be stated as: let X 1 , X 2 , ⊠, X n {\\displaystyle X\_{1},X\_{2},\\dots ,X\_{n}}  denote a [statistical sample](https://en.wikipedia.org/wiki/Sampling_\(statistics\) "Sampling (statistics)") of size n {\\displaystyle n}  from a population with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") (average) ÎŒ {\\displaystyle \\mu }  and finite positive [variance](https://en.wikipedia.org/wiki/Variance "Variance") Ï 2 {\\displaystyle \\sigma ^{2}} , and let X ÂŻ n {\\displaystyle {\\bar {X}}\_{n}}  denote the sample mean (which is itself a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable")). Then the [limit as n â â {\\displaystyle n\\to \\infty }  of the distribution](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_distribution "Convergence of random variables") of ( X ÂŻ n â ÎŒ ) n {\\displaystyle ({\\bar {X}}\_{n}-\\mu ){\\sqrt {n}}}  is a normal distribution with mean 0 {\\displaystyle 0}  and variance Ï 2 {\\displaystyle \\sigma ^{2}} .[\[2\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-2)
In other words, suppose that a large sample of [observations](https://en.wikipedia.org/wiki/Random_variate "Random variate") is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average ([arithmetic mean](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean")) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the [probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") of these averages will closely approximate a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be [independent and identically distributed](https://en.wikipedia.org/wiki/Independent_and_identically_distributed "Independent and identically distributed") (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution"), is the [de MoivreâLaplace theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem "De MoivreâLaplace theorem").
## Independent sequences
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=1 "Edit section: Independent sequences")\]
[](https://en.wikipedia.org/wiki/File:IllustrationCentralTheorem.png)
Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.[\[3\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-3)
### Classical CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=2 "Edit section: Classical CLT")\]
Let ( X n ) n â„ 1 {\\displaystyle (X\_{n})\_{n\\geq 1}}  be a sequence of [i.i.d. random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables") having a distribution with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") given by ÎŒ {\\displaystyle \\mu }  and finite [variance](https://en.wikipedia.org/wiki/Variance "Variance") given by Ï 2 . {\\displaystyle \\sigma ^{2}.}  Suppose we are interested in the [sample average](https://en.wikipedia.org/wiki/Sample_mean "Sample mean")
X ¯ n ⥠X 1 \+ ⯠\+ X n n . {\\displaystyle {\\bar {X}}\_{n}\\equiv {\\frac {X\_{1}+\\cdots +X\_{n}}{n}}.} 
By the [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers"), the sample average [converges almost surely](https://en.wikipedia.org/wiki/Almost_sure_convergence "Almost sure convergence") (and therefore also [converges in probability](https://en.wikipedia.org/wiki/Convergence_in_probability "Convergence in probability")) to the expected value ÎŒ {\\displaystyle \\mu }  as n â â . {\\displaystyle n\\to \\infty .} 
The classical central limit theorem describes the size and the distributional form of the [stochastic](https://en.wiktionary.org/wiki/stochastic "wikt:stochastic") fluctuations around the deterministic number ÎŒ {\\displaystyle \\mu }  during this convergence. More precisely, it states that as n {\\displaystyle n}  gets larger, the distribution of the normalized mean n ( X ÂŻ n â ÎŒ ) {\\displaystyle {\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )} , i.e. the difference between the sample average X ÂŻ n {\\displaystyle {\\bar {X}}\_{n}}  and its limit ÎŒ , {\\displaystyle \\mu ,}  scaled by the factor n {\\displaystyle {\\sqrt {n}}} , approaches the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with mean 0 {\\displaystyle 0}  and variance Ï 2 . {\\displaystyle \\sigma ^{2}.}  For large enough n , {\\displaystyle n,}  the distribution of X ÂŻ n {\\displaystyle {\\bar {X}}\_{n}}  gets arbitrarily close to the normal distribution with mean ÎŒ {\\displaystyle \\mu }  and variance Ï 2 / n . {\\displaystyle \\sigma ^{2}/n.} 
The usefulness of the theorem is that the distribution of n ( X ÂŻ n â ÎŒ ) {\\displaystyle {\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}  approaches normality regardless of the shape of the distribution of the individual X i . {\\displaystyle X\_{i}.}  Formally, the theorem can be stated as follows:
**LindebergâLĂ©vy CLT**âSuppose X 1 , X 2 , X 3 ⊠{\\displaystyle X\_{1},X\_{2},X\_{3}\\ldots }  is a sequence of [i.i.d.](https://en.wikipedia.org/wiki/Independent_and_identically_distributed "Independent and identically distributed") random variables with E ⥠\[ X i \] \= ÎŒ {\\displaystyle \\operatorname {E} \[X\_{i}\]=\\mu } ![{\\displaystyle \\operatorname {E} \[X\_{i}\]=\\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cb0f7a2cebcaf8b28056deff750e7a0e1d34f3) and Var ⥠\[ X i \] \= Ï 2 \< â . {\\displaystyle \\operatorname {Var} \[X\_{i}\]=\\sigma ^{2}\<\\infty .} ![{\\displaystyle \\operatorname {Var} \[X\_{i}\]=\\sigma ^{2}\<\\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76c2379b589ea26ef8d17aaeab2d9899f30f1299) Then, as n {\\displaystyle n}  approaches infinity, the random variables n ( X ÂŻ n â ÎŒ ) {\\displaystyle {\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}  [converge in distribution](https://en.wikipedia.org/wiki/Convergence_in_distribution "Convergence in distribution") to a [normal](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") N ( 0 , Ï 2 ) {\\displaystyle {\\mathcal {N}}(0,\\sigma ^{2})} :[\[4\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995357-4)
n ( X ÂŻ n â ÎŒ ) ⶠd N ( 0 , Ï 2 ) . {\\displaystyle {\\sqrt {n}}\\left({\\bar {X}}\_{n}-\\mu \\right)\\mathrel {\\overset {d}{\\longrightarrow }} {\\mathcal {N}}\\left(0,\\sigma ^{2}\\right).} 
In the case Ï \> 0 , {\\displaystyle \\sigma \>0,}  convergence in distribution means that the [cumulative distribution functions](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function") of n ( X ÂŻ n â ÎŒ ) {\\displaystyle {\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}  converge pointwise to the cdf of the N ( 0 , Ï 2 ) {\\displaystyle {\\mathcal {N}}(0,\\sigma ^{2})}  distribution: for every real number z , {\\displaystyle z,} 
lim n â â P \[ n ( X ÂŻ n â ÎŒ ) †z \] \= lim n â â P \[ n ( X ÂŻ n â ÎŒ ) Ï â€ z Ï \] \= Ί ( z Ï ) , {\\displaystyle \\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]=\\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\frac {{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}{\\sigma }}\\leq {\\frac {z}{\\sigma }}\\right\]=\\Phi \\left({\\frac {z}{\\sigma }}\\right),} ![{\\displaystyle \\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]=\\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\frac {{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}{\\sigma }}\\leq {\\frac {z}{\\sigma }}\\right\]=\\Phi \\left({\\frac {z}{\\sigma }}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/defd4cf70972fa6a76a8570fee6551f4cb7d70b8)
where Ί ( z ) {\\displaystyle \\Phi (z)}  is the standard normal cdf evaluated at z . {\\displaystyle z.}  The convergence is uniform in z {\\displaystyle z}  in the sense that
lim n â â sup z â R \| P \[ n ( X ÂŻ n â ÎŒ ) †z \] â Ί ( z Ï ) \| \= 0 , {\\displaystyle \\lim \_{n\\to \\infty }\\;\\sup \_{z\\in \\mathbb {R} }\\;\\left\|\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]-\\Phi \\left({\\frac {z}{\\sigma }}\\right)\\right\|=0~,} ![{\\displaystyle \\lim \_{n\\to \\infty }\\;\\sup \_{z\\in \\mathbb {R} }\\;\\left\|\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]-\\Phi \\left({\\frac {z}{\\sigma }}\\right)\\right\|=0~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835addcb3ec37594d1e9a6a78c0373a5e7b2eddc)
where sup {\\displaystyle \\sup }  denotes the [supremum](https://en.wikipedia.org/wiki/Supremum "Supremum") (i.e. least upper bound) of the set.[\[5\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBauer2001199Theorem_30.13-5)
### Lyapunov CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=3 "Edit section: Lyapunov CLT")\]
In this variant of the central limit theorem the random variables X i {\\textstyle X\_{i}}  have to be independent, but not necessarily identically distributed. The theorem also requires that random variables \| X i \| {\\textstyle \\left\|X\_{i}\\right\|}  have [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of some order ( 2 \+ ÎŽ ) {\\textstyle (2+\\delta )}  , and that the rate of growth of these moments is limited by the Lyapunov condition given below.
**Lyapunov CLT[\[6\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995362-6)**âSuppose { X 1 , ⊠, X n , ⊠} {\\textstyle \\{X\_{1},\\ldots ,X\_{n},\\ldots \\}}  is a sequence of independent random variables, each with finite expected value ÎŒ i {\\textstyle \\mu \_{i}}  and variance Ï i 2 {\\textstyle \\sigma \_{i}^{2}}  . Define
s n 2 \= â i \= 1 n Ï i 2 . {\\displaystyle s\_{n}^{2}=\\sum \_{i=1}^{n}\\sigma \_{i}^{2}.} 
If for some ÎŽ \> 0 {\\textstyle \\delta \>0}  , *Lyapunovâs condition*
lim n â â 1 s n 2 \+ ÎŽ â i \= 1 n E ⥠\[ \| X i â ÎŒ i \| 2 \+ ÎŽ \] \= 0 {\\displaystyle \\lim \_{n\\to \\infty }\\;{\\frac {1}{s\_{n}^{2+\\delta }}}\\,\\sum \_{i=1}^{n}\\operatorname {E} \\left\[\\left\|X\_{i}-\\mu \_{i}\\right\|^{2+\\delta }\\right\]=0} ![{\\displaystyle \\lim \_{n\\to \\infty }\\;{\\frac {1}{s\_{n}^{2+\\delta }}}\\,\\sum \_{i=1}^{n}\\operatorname {E} \\left\[\\left\|X\_{i}-\\mu \_{i}\\right\|^{2+\\delta }\\right\]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f06b7a3309fd45005cce7a4e0b15ca3758f662f5)
is satisfied, then a sum of X i â ÎŒ i s n {\\textstyle {\\frac {X\_{i}-\\mu \_{i}}{s\_{n}}}}  converges in distribution to a standard normal random variable, as n {\\textstyle n}  goes to infinity:
1 s n â i \= 1 n ( X i â ÎŒ i ) ⶠd N ( 0 , 1 ) . {\\displaystyle {\\frac {1}{s\_{n}}}\\,\\sum \_{i=1}^{n}\\left(X\_{i}-\\mu \_{i}\\right)\\mathrel {\\overset {d}{\\longrightarrow }} {\\mathcal {N}}(0,1).} 
In practice it is usually easiest to check Lyapunov's condition for ÎŽ \= 1 {\\textstyle \\delta =1}  .
If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
### Lindeberg (-Feller) CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=4 "Edit section: Lindeberg (-Feller) CLT")\]
Main article: [Lindeberg's condition](https://en.wikipedia.org/wiki/Lindeberg%27s_condition "Lindeberg's condition")
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from [Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg") in 1920).
Suppose that for every Δ \> 0 {\\textstyle \\varepsilon \>0} ,
lim n â â 1 s n 2 â i \= 1 n E ⥠\[ ( X i â ÎŒ i ) 2 â
1 { \| X i â ÎŒ i \| \> Δ s n } \] \= 0 {\\displaystyle \\lim \_{n\\to \\infty }{\\frac {1}{s\_{n}^{2}}}\\sum \_{i=1}^{n}\\operatorname {E} \\left\[(X\_{i}-\\mu \_{i})^{2}\\cdot \\mathbf {1} \_{\\left\\{\\left\|X\_{i}-\\mu \_{i}\\right\|\>\\varepsilon s\_{n}\\right\\}}\\right\]=0} ![{\\displaystyle \\lim \_{n\\to \\infty }{\\frac {1}{s\_{n}^{2}}}\\sum \_{i=1}^{n}\\operatorname {E} \\left\[(X\_{i}-\\mu \_{i})^{2}\\cdot \\mathbf {1} \_{\\left\\{\\left\|X\_{i}-\\mu \_{i}\\right\|\>\\varepsilon s\_{n}\\right\\}}\\right\]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd10d152dd578ac2a2fa674a084bd7b03b95b1b)
where 1 { ⊠} {\\textstyle \\mathbf {1} \_{\\{\\ldots \\}}}  is the [indicator function](https://en.wikipedia.org/wiki/Indicator_function "Indicator function"). Then the distribution of the standardized sums
1 s n â i \= 1 n ( X i â ÎŒ i ) {\\displaystyle {\\frac {1}{s\_{n}}}\\sum \_{i=1}^{n}\\left(X\_{i}-\\mu \_{i}\\right)} 
converges towards the standard normal distribution N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  .
### CLT for the sum of a random number of random variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=5 "Edit section: CLT for the sum of a random number of random variables")\]
Rather than summing an integer number n {\\displaystyle n}  of random variables and taking n â â {\\displaystyle n\\to \\infty } , the sum can be of a random number N {\\displaystyle N}  of random variables, with conditions on N {\\displaystyle N} . For example, the following theorem is Corollary 4 of Robbins (1948). It assumes that N {\\displaystyle N}  is asymptotically normal (Robbins also developed other conditions that lead to the same result).
**Robbins CLT[\[7\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-8)**âLet { X i , i â„ 1 } {\\displaystyle \\{X\_{i},i\\geq 1\\}}  be independent, identically distributed random variables with E ( X i ) \= ÎŒ {\\displaystyle E(X\_{i})=\\mu }  and Var ( X i ) \= Ï 2 {\\displaystyle {\\text{Var}}(X\_{i})=\\sigma ^{2}} , and let { N n , n â„ 1 } {\\displaystyle \\{N\_{n},n\\geq 1\\}}  be a sequence of non-negative integer-valued random variables that are independent of { X i , i â„ 1 } {\\displaystyle \\{X\_{i},i\\geq 1\\}} . Assume for each n \= 1 , 2 , ⊠{\\displaystyle n=1,2,\\dots }  that E ( N n 2 ) \< â {\\displaystyle E(N\_{n}^{2})\<\\infty }  and
N n â E ( N n ) Var ( N n ) â d N ( 0 , 1 ) {\\displaystyle {\\frac {N\_{n}-E(N\_{n})}{\\sqrt {{\\text{Var}}(N\_{n})}}}\\xrightarrow {\\quad d\\quad } {\\mathcal {N}}(0,1)} 
where â d {\\displaystyle \\xrightarrow {\\,d\\,} }  denotes convergence in distribution and N ( 0 , 1 ) {\\displaystyle {\\mathcal {N}}(0,1)}  is the normal distribution with mean 0, variance 1. Then
â i \= 1 N n X i â ÎŒ E ( N n ) Ï 2 E ( N n ) \+ ÎŒ 2 Var ( N n ) â d N ( 0 , 1 ) {\\displaystyle {\\frac {\\sum \_{i=1}^{N\_{n}}X\_{i}-\\mu E(N\_{n})}{\\sqrt {\\sigma ^{2}E(N\_{n})+\\mu ^{2}{\\text{Var}}(N\_{n})}}}\\xrightarrow {\\quad d\\quad } {\\mathcal {N}}(0,1)} 
### Multidimensional CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=6 "Edit section: Multidimensional CLT")\]
Proofs that use characteristic functions can be extended to cases where each individual X i {\\textstyle \\mathbf {X} \_{i}}  is a [random vector](https://en.wikipedia.org/wiki/Random_vector "Random vector") in R k {\\textstyle \\mathbb {R} ^{k}}  , with mean vector ÎŒ \= E ⥠\[ X i \] {\\textstyle {\\boldsymbol {\\mu }}=\\operatorname {E} \[\\mathbf {X} \_{i}\]} ![{\\textstyle {\\boldsymbol {\\mu }}=\\operatorname {E} \[\\mathbf {X} \_{i}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2931b016ce578d17578ee3cdffeb31852446873) and [covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix "Covariance matrix") ÎŁ {\\textstyle \\mathbf {\\Sigma } }  (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a [multivariate normal distribution](https://en.wikipedia.org/wiki/Multivariate_normal_distribution "Multivariate normal distribution").[\[9\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-vanderVaart-9) Summation of these vectors is done component-wise.
For i \= 1 , 2 , 3 , ⊠, {\\displaystyle i=1,2,3,\\ldots ,}  let
X i \= \[ X i ( 1 ) âź X i ( k ) \] {\\displaystyle \\mathbf {X} \_{i}={\\begin{bmatrix}X\_{i}^{(1)}\\\\\\vdots \\\\X\_{i}^{(k)}\\end{bmatrix}}} 
be independent random vectors. The sum of the random vectors X 1 , ⊠, X n {\\displaystyle \\mathbf {X} \_{1},\\ldots ,\\mathbf {X} \_{n}}  is
â i \= 1 n X i \= \[ X 1 ( 1 ) âź X 1 ( k ) \] \+ \[ X 2 ( 1 ) âź X 2 ( k ) \] \+ ⯠\+ \[ X n ( 1 ) âź X n ( k ) \] \= \[ â i \= 1 n X i ( 1 ) âź â i \= 1 n X i ( k ) \] {\\displaystyle \\sum \_{i=1}^{n}\\mathbf {X} \_{i}={\\begin{bmatrix}X\_{1}^{(1)}\\\\\\vdots \\\\X\_{1}^{(k)}\\end{bmatrix}}+{\\begin{bmatrix}X\_{2}^{(1)}\\\\\\vdots \\\\X\_{2}^{(k)}\\end{bmatrix}}+\\cdots +{\\begin{bmatrix}X\_{n}^{(1)}\\\\\\vdots \\\\X\_{n}^{(k)}\\end{bmatrix}}={\\begin{bmatrix}\\sum \_{i=1}^{n}X\_{i}^{(1)}\\\\\\vdots \\\\\\sum \_{i=1}^{n}X\_{i}^{(k)}\\end{bmatrix}}} 
and their average is
X ÂŻ n \= \[ X ÂŻ i ( 1 ) âź X ÂŻ i ( k ) \] \= 1 n â i \= 1 n X i . {\\displaystyle \\mathbf {{\\bar {X}}\_{n}} ={\\begin{bmatrix}{\\bar {X}}\_{i}^{(1)}\\\\\\vdots \\\\{\\bar {X}}\_{i}^{(k)}\\end{bmatrix}}={\\frac {1}{n}}\\sum \_{i=1}^{n}\\mathbf {X} \_{i}.} 
Therefore,
1 n â i \= 1 n \[ X i â E ⥠( X i ) \] \= 1 n â i \= 1 n ( X i â ÎŒ ) \= n ( X ÂŻ n â ÎŒ ) . {\\displaystyle {\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}\\left\[\\mathbf {X} \_{i}-\\operatorname {E} \\left(\\mathbf {X} \_{i}\\right)\\right\]={\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}(\\mathbf {X} \_{i}-{\\boldsymbol {\\mu }})={\\sqrt {n}}\\left({\\overline {\\mathbf {X} }}\_{n}-{\\boldsymbol {\\mu }}\\right).} ![{\\displaystyle {\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}\\left\[\\mathbf {X} \_{i}-\\operatorname {E} \\left(\\mathbf {X} \_{i}\\right)\\right\]={\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}(\\mathbf {X} \_{i}-{\\boldsymbol {\\mu }})={\\sqrt {n}}\\left({\\overline {\\mathbf {X} }}\_{n}-{\\boldsymbol {\\mu }}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d3c26b99d8a97339b688367c519d0e28f9541c)
The multivariate central limit theorem states that
n ( X ÂŻ n â ÎŒ ) ⶠd N k ( 0 , ÎŁ ) , {\\displaystyle {\\sqrt {n}}\\left({\\overline {\\mathbf {X} }}\_{n}-{\\boldsymbol {\\mu }}\\right)\\mathrel {\\overset {d}{\\longrightarrow }} {\\mathcal {N}}\_{k}(0,{\\boldsymbol {\\Sigma }}),}  where the [covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix "Covariance matrix") ÎŁ {\\displaystyle {\\boldsymbol {\\Sigma }}}  is equal to ÎŁ \= \[ Var ⥠( X 1 ( 1 ) ) Cov ⥠( X 1 ( 1 ) , X 1 ( 2 ) ) Cov ⥠( X 1 ( 1 ) , X 1 ( 3 ) ) ⯠Cov ⥠( X 1 ( 1 ) , X 1 ( k ) ) Cov ⥠( X 1 ( 2 ) , X 1 ( 1 ) ) Var ⥠( X 1 ( 2 ) ) Cov ⥠( X 1 ( 2 ) , X 1 ( 3 ) ) ⯠Cov ⥠( X 1 ( 2 ) , X 1 ( k ) ) Cov ⥠( X 1 ( 3 ) , X 1 ( 1 ) ) Cov ⥠( X 1 ( 3 ) , X 1 ( 2 ) ) Var ⥠( X 1 ( 3 ) ) ⯠Cov ⥠( X 1 ( 3 ) , X 1 ( k ) ) âź âź âź â± âź Cov ⥠( X 1 ( k ) , X 1 ( 1 ) ) Cov ⥠( X 1 ( k ) , X 1 ( 2 ) ) Cov ⥠( X 1 ( k ) , X 1 ( 3 ) ) ⯠Var ⥠( X 1 ( k ) ) \] . {\\displaystyle {\\boldsymbol {\\Sigma }}={\\begin{bmatrix}{\\operatorname {Var} \\left(X\_{1}^{(1)}\\right)}&\\operatorname {Cov} \\left(X\_{1}^{(1)},X\_{1}^{(2)}\\right)&\\operatorname {Cov} \\left(X\_{1}^{(1)},X\_{1}^{(3)}\\right)&\\cdots &\\operatorname {Cov} \\left(X\_{1}^{(1)},X\_{1}^{(k)}\\right)\\\\\\operatorname {Cov} \\left(X\_{1}^{(2)},X\_{1}^{(1)}\\right)&\\operatorname {Var} \\left(X\_{1}^{(2)}\\right)&\\operatorname {Cov} \\left(X\_{1}^{(2)},X\_{1}^{(3)}\\right)&\\cdots &\\operatorname {Cov} \\left(X\_{1}^{(2)},X\_{1}^{(k)}\\right)\\\\\\operatorname {Cov} \\left(X\_{1}^{(3)},X\_{1}^{(1)}\\right)&\\operatorname {Cov} \\left(X\_{1}^{(3)},X\_{1}^{(2)}\\right)&\\operatorname {Var} \\left(X\_{1}^{(3)}\\right)&\\cdots &\\operatorname {Cov} \\left(X\_{1}^{(3)},X\_{1}^{(k)}\\right)\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\\\operatorname {Cov} \\left(X\_{1}^{(k)},X\_{1}^{(1)}\\right)&\\operatorname {Cov} \\left(X\_{1}^{(k)},X\_{1}^{(2)}\\right)&\\operatorname {Cov} \\left(X\_{1}^{(k)},X\_{1}^{(3)}\\right)&\\cdots &\\operatorname {Var} \\left(X\_{1}^{(k)}\\right)\\\\\\end{bmatrix}}~.} 
The multivariate central limit theorem can be proved using the [CramĂ©râWold theorem](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Wold_theorem "CramĂ©râWold theorem").[\[9\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-vanderVaart-9)
The rate of convergence is given by the following [BerryâEsseen](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem") type result:
**Theorem[\[10\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-10)**âLet X 1 , ⊠, X n , ⊠{\\displaystyle X\_{1},\\dots ,X\_{n},\\dots }  be independent R d {\\displaystyle \\mathbb {R} ^{d}} \-valued random vectors, each having mean zero. Write S \= â i \= 1 n X i {\\displaystyle S=\\sum \_{i=1}^{n}X\_{i}}  and assume ÎŁ \= Cov ⥠\[ S \] {\\displaystyle \\Sigma =\\operatorname {Cov} \[S\]} ![{\\displaystyle \\Sigma =\\operatorname {Cov} \[S\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbdd46ff8928b02fc4e37a7fc14c51ceaf58b40) is invertible. Let Z ⌠N ( 0 , ÎŁ ) {\\displaystyle Z\\sim {\\mathcal {N}}(0,\\Sigma )}  be a d {\\displaystyle d} \-dimensional Gaussian with the same mean and same covariance matrix as S {\\displaystyle S} . Then for all convex sets U â R d {\\displaystyle U\\subseteq \\mathbb {R} ^{d}}  ,
\| P \[ S â U \] â P \[ Z â U \] \| †C d 1 / 4 Îł , {\\displaystyle \\left\|\\mathbb {P} \[S\\in U\]-\\mathbb {P} \[Z\\in U\]\\right\|\\leq C\\,d^{1/4}\\gamma ~,} ![{\\displaystyle \\left\|\\mathbb {P} \[S\\in U\]-\\mathbb {P} \[Z\\in U\]\\right\|\\leq C\\,d^{1/4}\\gamma ~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c26c26cb1dedbb0db401fd2ebfb479ec45fb4cc) where C {\\displaystyle C}  is a universal constant, Îł \= â i \= 1 n E ⥠\[ â ÎŁ â 1 / 2 X i â 2 3 \] {\\displaystyle \\gamma =\\sum \_{i=1}^{n}\\operatorname {E} \\left\[\\left\\\|\\Sigma ^{-1/2}X\_{i}\\right\\\|\_{2}^{3}\\right\]} ![{\\displaystyle \\gamma =\\sum \_{i=1}^{n}\\operatorname {E} \\left\[\\left\\\|\\Sigma ^{-1/2}X\_{i}\\right\\\|\_{2}^{3}\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7904b82b37828200b3a729334fb37932175ef82e) , and â â
â 2 {\\displaystyle \\\|\\cdot \\\|\_{2}}  denotes the Euclidean norm on R d {\\displaystyle \\mathbb {R} ^{d}}  .
It is unknown whether the factor d 1 / 4 {\\textstyle d^{1/4}}  is necessary.[\[11\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-11)
## The generalized central limit theorem
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=7 "Edit section: The generalized central limit theorem")\]
The generalized central limit theorem (GCLT) was an effort of multiple mathematicians ([Sergei Bernstein](https://en.wikipedia.org/wiki/Sergei_Bernstein "Sergei Bernstein"), [Jarl Waldemar Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg"), [Paul LĂ©vy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul LĂ©vy (mathematician)"), [William Feller](https://en.wikipedia.org/wiki/William_Feller "William Feller"), [Andrey Kolmogorov](https://en.wikipedia.org/wiki/Andrey_Kolmogorov "Andrey Kolmogorov"), and others) over the period from 1920 to 1937.[\[12\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-12) The first published complete proof of the GCLT was in 1937 by Paul LĂ©vy in French.[\[13\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-13) An English language version of the complete proof of the GCLT is available in the translation of [Boris Vladimirovich Gnedenko](https://en.wikipedia.org/wiki/Boris_Vladimirovich_Gnedenko "Boris Vladimirovich Gnedenko") and Kolmogorov's 1954 book.[\[14\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-14)
The statement of the GCLT is as follows:[\[15\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-15)
**Statement of GCLT**âA non-degenerate random variable Z is [α\-stable](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution") for some 0 \< *α* †2 if and only if there is an independent, identically distributed sequence of random variables *X*1, *X*2, *X*3, ..., and constants *a**n* \> 0, *b**n* â â with a n ( X 1 \+ ⯠\+ X n ) â b n â Z . {\\displaystyle a\_{n}(X\_{1}+\\dots +X\_{n})-b\_{n}\\to Z.}  Here, 'â' means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy *F**n*(*y*) â *F*(*y*) at all continuity points of F.
In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a [stable distribution](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution").
## Dependent processes
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=8 "Edit section: Dependent processes")\]
### CLT under weak dependence
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=9 "Edit section: CLT under weak dependence")\]
A useful generalization of a sequence of independent, identically distributed random variables is a [mixing](https://en.wikipedia.org/wiki/Mixing_\(mathematics\) "Mixing (mathematics)") random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially [strong mixing](https://en.wikipedia.org/wiki/Mixing_\(mathematics\)#Mixing_in_stochastic_processes "Mixing (mathematics)") (also called α-mixing) defined by α ( n ) â 0 {\\textstyle \\alpha (n)\\to 0}  where α ( n ) {\\textstyle \\alpha (n)}  is so-called [strong mixing coefficient](https://en.wikipedia.org/wiki/Mixing_\(mathematics\)#Mixing_in_stochastic_processes "Mixing (mathematics)").
A simplified formulation of the central limit theorem under strong mixing is:[\[16\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995Theorem_27.4-16)
**Theorem**âSuppose that { X 1 , ⊠, X n , ⊠} {\\textstyle \\{X\_{1},\\ldots ,X\_{n},\\ldots \\}}  is stationary and α {\\displaystyle \\alpha } \-mixing with α n \= O ( n â 5 ) {\\textstyle \\alpha \_{n}=O\\left(n^{-5}\\right)}  and that E ⥠\[ X n \] \= 0 {\\textstyle \\operatorname {E} \[X\_{n}\]=0} ![{\\textstyle \\operatorname {E} \[X\_{n}\]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12315de3945900a1cdcca84088a0f562e93d042) and E ⥠\[ X n 12 \] \< â {\\textstyle \\operatorname {E} \[X\_{n}^{12}\]\<\\infty } ![{\\textstyle \\operatorname {E} \[X\_{n}^{12}\]\<\\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa251162abe2eb1e886116a08905a2e1f30ba891) . Denote S n \= X 1 \+ ⯠\+ X n {\\textstyle S\_{n}=X\_{1}+\\cdots +X\_{n}}  , then the limit
Ï 2 \= lim n â â E ⥠( S n 2 ) n {\\displaystyle \\sigma ^{2}=\\lim \_{n\\rightarrow \\infty }{\\frac {\\operatorname {E} \\left(S\_{n}^{2}\\right)}{n}}} 
exists, and if Ï â 0 {\\textstyle \\sigma \\neq 0}  then S n Ï n {\\textstyle {\\frac {S\_{n}}{\\sigma {\\sqrt {n}}}}}  converges in distribution to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)} .
In fact,
Ï 2 \= E ⥠( X 1 2 ) \+ 2 â k \= 1 â E ⥠( X 1 X 1 \+ k ) , {\\displaystyle \\sigma ^{2}=\\operatorname {E} \\left(X\_{1}^{2}\\right)+2\\sum \_{k=1}^{\\infty }\\operatorname {E} \\left(X\_{1}X\_{1+k}\\right),} 
where the series converges absolutely.
The assumption Ï â 0 {\\textstyle \\sigma \\neq 0}  cannot be omitted, since the asymptotic normality fails for X n \= Y n â Y n â 1 {\\textstyle X\_{n}=Y\_{n}-Y\_{n-1}}  where Y n {\\textstyle Y\_{n}}  are another [stationary sequence](https://en.wikipedia.org/wiki/Stationary_sequence "Stationary sequence").
There is a stronger version of the theorem:[\[17\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Sect._7.7\(c\),_Theorem_7.8-17) the assumption E ⥠\[ X n 12 \] \< â {\\textstyle \\operatorname {E} \\left\[X\_{n}^{12}\\right\]\<\\infty } ![{\\textstyle \\operatorname {E} \\left\[X\_{n}^{12}\\right\]\<\\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/657baea1ae3ea930c0d5057a69e2353a155a0df4) is replaced with E ⥠\[ \| X n \| 2 \+ ÎŽ \] \< â {\\textstyle \\operatorname {E} \\left\[{\\left\|X\_{n}\\right\|}^{2+\\delta }\\right\]\<\\infty } ![{\\textstyle \\operatorname {E} \\left\[{\\left\|X\_{n}\\right\|}^{2+\\delta }\\right\]\<\\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/940680c6ace3f0b2a962e5e31fcc79b6e4f28f13) , and the assumption α n \= O ( n â 5 ) {\\textstyle \\alpha \_{n}=O\\left(n^{-5}\\right)}  is replaced with
â n α n ÎŽ 2 ( 2 \+ ÎŽ ) \< â . {\\displaystyle \\sum \_{n}\\alpha \_{n}^{\\frac {\\delta }{2(2+\\delta )}}\<\\infty .} 
Existence of such ÎŽ \> 0 {\\textstyle \\delta \>0}  ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see ([Bradley 2007](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBradley2007)).
### Martingale difference CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=10 "Edit section: Martingale difference CLT")\]
Main article: [Martingale central limit theorem](https://en.wikipedia.org/wiki/Martingale_central_limit_theorem "Martingale central limit theorem")
**Theorem**âLet a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") M n {\\textstyle M\_{n}}  satisfy
- 1
n
â
k
\=
1
n
E
âĄ
\[
(
M
k
â
M
k
â
1
)
2
âŁ
M
1
,
âŠ
,
M
k
â
1
\]
â
1
{\\displaystyle {\\frac {1}{n}}\\sum \_{k=1}^{n}\\operatorname {E} \\left\[\\left(M\_{k}-M\_{k-1}\\right)^{2}\\mid M\_{1},\\dots ,M\_{k-1}\\right\]\\to 1}
![{\\displaystyle {\\frac {1}{n}}\\sum \_{k=1}^{n}\\operatorname {E} \\left\[\\left(M\_{k}-M\_{k-1}\\right)^{2}\\mid M\_{1},\\dots ,M\_{k-1}\\right\]\\to 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fc9c003077c427537a3a787e84ef5528bbe866)
in probability as *n* â â,
- for every *Δ* \> 0,
1
n
â
k
\=
1
n
E
âĄ
\[
(
M
k
â
M
k
â
1
)
2
1
\[
\|
M
k
â
M
k
â
1
\|
\>
Δ
n
\]
\]
â
0
{\\displaystyle {\\frac {1}{n}}\\sum \_{k=1}^{n}{\\operatorname {E} \\left\[\\left(M\_{k}-M\_{k-1}\\right)^{2}\\mathbf {1} \\left\[\|M\_{k}-M\_{k-1}\|\>\\varepsilon {\\sqrt {n}}\\right\]\\right\]}\\to 0}
![{\\displaystyle {\\frac {1}{n}}\\sum \_{k=1}^{n}{\\operatorname {E} \\left\[\\left(M\_{k}-M\_{k-1}\\right)^{2}\\mathbf {1} \\left\[\|M\_{k}-M\_{k-1}\|\>\\varepsilon {\\sqrt {n}}\\right\]\\right\]}\\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a52a1cd0940afa0a6752d55edc28d1c72ab1bd)
as *n* â â,
then M n n {\\textstyle {\\frac {M\_{n}}{\\sqrt {n}}}}  converges in distribution to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  as n â â {\\textstyle n\\to \\infty } .[\[18\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4-18)[\[19\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995Theorem_35.12-19)
## Remarks
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=11 "Edit section: Remarks")\]
### Proof of classical CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=12 "Edit section: Proof of classical CLT")\]
The central limit theorem has a proof using [characteristic functions](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)").[\[20\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-20) It is similar to the proof of the (weak) [law of large numbers](https://en.wikipedia.org/wiki/Proof_of_the_law_of_large_numbers "Proof of the law of large numbers").
Assume { X 1 , ⊠, X n , ⊠} {\\textstyle \\{X\_{1},\\ldots ,X\_{n},\\ldots \\}}  are independent and identically distributed random variables, each with mean ÎŒ {\\textstyle \\mu }  and finite variance Ï 2 {\\textstyle \\sigma ^{2}}  . The sum X 1 \+ ⯠\+ X n {\\textstyle X\_{1}+\\cdots +X\_{n}}  has [mean](https://en.wikipedia.org/wiki/Linearity_of_expectation "Linearity of expectation") n ÎŒ {\\textstyle n\\mu }  and [variance](https://en.wikipedia.org/wiki/Variance#Sum_of_uncorrelated_variables_\(Bienaym%C3%A9_formula\) "Variance") n Ï 2 {\\textstyle n\\sigma ^{2}}  . Consider the random variable
Z n \= X 1 \+ ⯠\+ X n â n ÎŒ n Ï 2 \= â i \= 1 n X i â ÎŒ n Ï 2 \= â i \= 1 n 1 n Y i , {\\displaystyle Z\_{n}={\\frac {X\_{1}+\\cdots +X\_{n}-n\\mu }{\\sqrt {n\\sigma ^{2}}}}=\\sum \_{i=1}^{n}{\\frac {X\_{i}-\\mu }{\\sqrt {n\\sigma ^{2}}}}=\\sum \_{i=1}^{n}{\\frac {1}{\\sqrt {n}}}Y\_{i},} 
where in the last step we defined the new random variables Y i \= X i â ÎŒ Ï {\\textstyle Y\_{i}={\\frac {X\_{i}-\\mu }{\\sigma }}}  , each with zero mean and unit variance ( var ⥠( Y ) \= 1 {\\textstyle \\operatorname {var} (Y)=1}  ). The [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)") of Z n {\\textstyle Z\_{n}}  is given by
Ï Z n ( t ) \= Ï â i \= 1 n 1 n Y i ( t ) \= Ï Y 1 ( t n ) Ï Y 2 ( t n ) âŻ Ï Y n ( t n ) \= \[ Ï Y 1 ( t n ) \] n , {\\displaystyle {\\begin{aligned}\\varphi \_{Z\_{n}}\\!(t)=\\varphi \_{\\sum \_{i=1}^{n}{{\\frac {1}{\\sqrt {n}}}Y\_{i}}}\\!(t)\\ &=\\ \\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\varphi \_{Y\_{2}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\cdots \\varphi \_{Y\_{n}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\\\\[1ex\]&=\\ \\left\[\\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\right\]^{n},\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\varphi \_{Z\_{n}}\\!(t)=\\varphi \_{\\sum \_{i=1}^{n}{{\\frac {1}{\\sqrt {n}}}Y\_{i}}}\\!(t)\\ &=\\ \\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\varphi \_{Y\_{2}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\cdots \\varphi \_{Y\_{n}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\\\\[1ex\]&=\\ \\left\[\\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\right\]^{n},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b0081e06952eb33ecd2d847e4c8bf6c41449bd)
where in the last step we used the fact that all of the Y i {\\textstyle Y\_{i}}  are identically distributed. The characteristic function of Y 1 {\\textstyle Y\_{1}}  is, by [Taylor's theorem](https://en.wikipedia.org/wiki/Taylor%27s_theorem "Taylor's theorem"), Ï Y 1 ( t n ) \= 1 â t 2 2 n \+ o ( t 2 n ) , ( t n ) â 0 {\\displaystyle \\varphi \_{Y\_{1}}\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)=1-{\\frac {t^{2}}{2n}}+o\\!\\left({\\frac {t^{2}}{n}}\\right),\\quad \\left({\\frac {t}{\\sqrt {n}}}\\right)\\to 0} 
where o ( t 2 / n ) {\\textstyle o(t^{2}/n)}  is "[little o notation](https://en.wikipedia.org/wiki/Little-o_notation "Little-o notation")" for some function of t {\\textstyle t}  that goes to zero more rapidly than t 2 / n {\\textstyle t^{2}/n}  . By the limit of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") ( e x \= lim n â â ( 1 \+ x n ) n {\\textstyle e^{x}=\\lim \_{n\\to \\infty }\\left(1+{\\frac {x}{n}}\\right)^{n}}  ), the characteristic function of Z n {\\displaystyle Z\_{n}}  equals
Ï Z n ( t ) \= ( 1 â t 2 2 n \+ o ( t 2 n ) ) n â e â 1 2 t 2 , n â â . {\\displaystyle \\varphi \_{Z\_{n}}(t)=\\left(1-{\\frac {t^{2}}{2n}}+o\\left({\\frac {t^{2}}{n}}\\right)\\right)^{n}\\rightarrow e^{-{\\frac {1}{2}}t^{2}},\\quad n\\to \\infty .} 
All of the higher order terms vanish in the limit n â â {\\textstyle n\\to \\infty }  . The right hand side equals the characteristic function of a standard normal distribution N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)} , which implies through [LĂ©vy's continuity theorem](https://en.wikipedia.org/wiki/L%C3%A9vy_continuity_theorem "LĂ©vy continuity theorem") that the distribution of Z n {\\textstyle Z\_{n}}  will approach N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  as n â â {\\textstyle n\\to \\infty }  . Therefore, the [sample average](https://en.wikipedia.org/wiki/Sample_mean "Sample mean")
X ¯ n \= X 1 \+ ⯠\+ X n n {\\displaystyle {\\bar {X}}\_{n}={\\frac {X\_{1}+\\cdots +X\_{n}}{n}}} 
is such that
n Ï ( X ÂŻ n â ÎŒ ) \= Z n {\\displaystyle {\\frac {\\sqrt {n}}{\\sigma }}\\left({\\bar {X}}\_{n}-\\mu \\right)=Z\_{n}} 
converges to the normal distribution N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  , from which the central limit theorem follows.
### Convergence to the limit
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=13 "Edit section: Convergence to the limit")\]
The central limit theorem gives only an [asymptotic distribution](https://en.wikipedia.org/wiki/Asymptotic_distribution "Asymptotic distribution"). As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The convergence in the central limit theorem is [uniform](https://en.wikipedia.org/wiki/Uniform_convergence "Uniform convergence") because the limiting cumulative distribution function is continuous. If the third central [moment](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") E ⥠\[ ( X 1 â ÎŒ ) 3 \] {\\textstyle \\operatorname {E} \\left\[(X\_{1}-\\mu )^{3}\\right\]} ![{\\textstyle \\operatorname {E} \\left\[(X\_{1}-\\mu )^{3}\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389b355cd56db15cdf7e88c8b0aff830a381726f) exists and is finite, then the speed of convergence is at least on the order of 1 / n {\\textstyle 1/{\\sqrt {n}}}  (see [BerryâEsseen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem")). [Stein's method](https://en.wikipedia.org/wiki/Stein%27s_method "Stein's method")[\[21\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-stein1972-21) can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.[\[22\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-22)
The convergence to the normal distribution is monotonic, in the sense that the [entropy](https://en.wikipedia.org/wiki/Information_entropy "Information entropy") of Z n {\\textstyle Z\_{n}}  increases [monotonically](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") to that of the normal distribution.[\[23\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-ABBN-23)
The central limit theorem applies in particular to sums of independent and identically distributed [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable"). A sum of [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable") is still a [discrete random variable](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable"), so that we are confronted with a sequence of [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable") whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution")). This means that if we build a [histogram](https://en.wikipedia.org/wiki/Histogram "Histogram") of the realizations of the sum of n independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity; this relation is known as [de MoivreâLaplace theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem "De MoivreâLaplace theorem"). The [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution") article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
### Common misconceptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=14 "Edit section: Common misconceptions")\]
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.[\[24\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-26) These include:
- The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of [iid](https://en.wikipedia.org/wiki/Iid "Iid") random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a [sampling distribution](https://en.wikipedia.org/wiki/Sampling_distribution "Sampling distribution") formed from different values of means (or sums) of such random variables.
- The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the [GlivenkoâCantelli theorem](https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem "GlivenkoâCantelli theorem").
- The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,[\[27\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-27) allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See [Z-test](https://en.wikipedia.org/wiki/Z-test "Z-test") for where the approximation holds.
### Relation to the law of large numbers
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=15 "Edit section: Relation to the law of large numbers")\]
The [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers") as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of Sn as n approaches infinity?" In mathematical analysis, [asymptotic series](https://en.wikipedia.org/wiki/Asymptotic_series "Asymptotic series") are one of the most popular tools employed to approach such questions.
Suppose we have an asymptotic expansion of f ( n ) {\\textstyle f(n)} :
f ( n ) \= a 1 Ï 1 ( n ) \+ a 2 Ï 2 ( n ) \+ O ( Ï 3 ( n ) ) ( n â â ) . {\\displaystyle f(n)=a\_{1}\\varphi \_{1}(n)+a\_{2}\\varphi \_{2}(n)+O{\\big (}\\varphi \_{3}(n){\\big )}\\qquad (n\\to \\infty ).} 
Dividing both parts by *Ï*1(*n*) and taking the limit will produce *a*1, the coefficient of the highest-order term in the expansion, which represents the rate at which *f*(*n*) changes in its leading term.
lim n â â f ( n ) Ï 1 ( n ) \= a 1 . {\\displaystyle \\lim \_{n\\to \\infty }{\\frac {f(n)}{\\varphi \_{1}(n)}}=a\_{1}.} 
Informally, one can say: "*f*(*n*) grows approximately as *a*1*Ï*1(*n*)". Taking the difference between *f*(*n*) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about *f*(*n*):
lim n â â f ( n ) â a 1 Ï 1 ( n ) Ï 2 ( n ) \= a 2 . {\\displaystyle \\lim \_{n\\to \\infty }{\\frac {f(n)-a\_{1}\\varphi \_{1}(n)}{\\varphi \_{2}(n)}}=a\_{2}.} 
Here one can say that the difference between the function and its approximation grows approximately as *a*2*Ï*2(*n*). The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.
Informally, something along these lines happens when the sum, Sn, of independent identically distributed random variables, *X*1, ..., *Xn*, is studied in classical probability theory.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] If each Xi has finite mean ÎŒ, then by the law of large numbers, â *Sn*/*n*â â *ÎŒ*.[\[28\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-28) If in addition each Xi has finite variance *Ï*2, then by the central limit theorem,
S n â n ÎŒ n â Ο , {\\displaystyle {\\frac {S\_{n}-n\\mu }{\\sqrt {n}}}\\to \\xi ,} 
where Ο is distributed as *N*(0,*Ï*2). This provides values of the first two constants in the informal expansion
S n â ÎŒ n \+ Ο n . {\\displaystyle S\_{n}\\approx \\mu n+\\xi {\\sqrt {n}}.} 
In the case where the Xi do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
S n â a n b n â Î , {\\displaystyle {\\frac {S\_{n}-a\_{n}}{b\_{n}}}\\rightarrow \\Xi ,} 
or informally
S n â a n \+ Î b n . {\\displaystyle S\_{n}\\approx a\_{n}+\\Xi b\_{n}.} 
Distributions Î which can arise in this way are called *[stable](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution")*.[\[29\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-29) Clearly, the normal distribution is stable, but there are also other stable distributions, such as the [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution "Cauchy distribution"), for which the mean or variance are not defined. The scaling factor bn may be proportional to nc, for any *c* â„ â 1/2â ; it may also be multiplied by a [slowly varying function](https://en.wikipedia.org/wiki/Slowly_varying_function "Slowly varying function") of n.[\[30\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Uchaikin-30)[\[31\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-31)
The [law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm "Law of the iterated logarithm") specifies what is happening "in between" the [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers") and the central limit theorem. Specifically it says that the normalizing function â*n* log log *n*, intermediate in size between n of the law of large numbers and â*n* of the central limit theorem, provides a non-trivial limiting behavior.
### Alternative statements of the theorem
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=16 "Edit section: Alternative statements of the theorem")\]
#### Density functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=17 "Edit section: Density functions")\]
The [density](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") of the sum of two or more independent variables is the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov[\[32\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-32) for a particular local limit theorem for sums of [independent and identically distributed random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables").
#### Characteristic functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=18 "Edit section: Characteristic functions")\]
Since the [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)") of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
An equivalent statement can be made about [Fourier transforms](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform"), since the characteristic function is essentially a Fourier transform.
### Calculating the variance
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=19 "Edit section: Calculating the variance")\]
Let Sn be the sum of n random variables. Many central limit theorems provide conditions such that Sn/âVar(Sn) converges in distribution to *N*(0,1) (the normal distribution with mean 0, variance 1) as n â â. In some cases, it is possible to find a constant *Ï*2 and function f(n) such that Sn/(Ïânâ
f(n)) converges in distribution to *N*(0,1) as nâ â.
**Lemma[\[33\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-33)**âSuppose X 1 , X 2 , ⊠{\\displaystyle X\_{1},X\_{2},\\dots }  is a sequence of real-valued and strictly stationary random variables with E ⥠( X i ) \= 0 {\\displaystyle \\operatorname {E} (X\_{i})=0}  for all i {\\displaystyle i}  , g : \[ 0 , 1 \] â R {\\displaystyle g:\[0,1\]\\to \\mathbb {R} } ![{\\displaystyle g:\[0,1\]\\to \\mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/26aef45c20ce13e8d53e79e068df9b5804c5c170) , and S n \= â i \= 1 n g ( i n ) X i {\\displaystyle S\_{n}=\\sum \_{i=1}^{n}g\\left({\\tfrac {i}{n}}\\right)X\_{i}}  . Construct
Ï 2 \= E ⥠( X 1 2 ) \+ 2 â i \= 1 â E ⥠( X 1 X 1 \+ i ) {\\displaystyle \\sigma ^{2}=\\operatorname {E} (X\_{1}^{2})+2\\sum \_{i=1}^{\\infty }\\operatorname {E} (X\_{1}X\_{1+i})} 
1. If
â
i
\=
1
â
E
âĄ
(
X
1
X
1
\+
i
)
{\\displaystyle \\sum \_{i=1}^{\\infty }\\operatorname {E} (X\_{1}X\_{1+i})}
is absolutely convergent,
\|
â«
0
1
g
(
x
)
g
âČ
(
x
)
d
x
\|
\<
â
{\\displaystyle \\left\|\\int \_{0}^{1}g(x)g'(x)\\,dx\\right\|\<\\infty }
, and
0
\<
â«
0
1
(
g
(
x
)
)
2
d
x
\<
â
{\\displaystyle 0\<\\int \_{0}^{1}(g(x))^{2}dx\<\\infty }
then
V
a
r
(
S
n
)
/
(
n
Îł
n
)
â
Ï
2
{\\displaystyle \\mathrm {Var} (S\_{n})/(n\\gamma \_{n})\\to \\sigma ^{2}}
as
n
â
â
{\\displaystyle n\\to \\infty }
where
Îł
n
\=
1
n
â
i
\=
1
n
(
g
(
i
n
)
)
2
{\\displaystyle \\gamma \_{n}={\\frac {1}{n}}\\sum \_{i=1}^{n}\\left(g\\left({\\tfrac {i}{n}}\\right)\\right)^{2}}
.
2. If in addition
Ï
\>
0
{\\displaystyle \\sigma \>0}
and
S
n
/
V
a
r
(
S
n
)
{\\displaystyle S\_{n}/{\\sqrt {\\mathrm {Var} (S\_{n})}}}
converges in distribution to
N
(
0
,
1
)
{\\displaystyle {\\mathcal {N}}(0,1)}
as
n
â
â
{\\displaystyle n\\to \\infty }
then
S
n
/
(
Ï
n
Îł
n
)
{\\displaystyle S\_{n}/(\\sigma {\\sqrt {n\\gamma \_{n}}})}
also converges in distribution to
N
(
0
,
1
)
{\\displaystyle {\\mathcal {N}}(0,1)}
as
n
â
â
{\\displaystyle n\\to \\infty }
.
## Extensions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=20 "Edit section: Extensions")\]
### Products of positive random variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=21 "Edit section: Products of positive random variables")\]
The [logarithm](https://en.wikipedia.org/wiki/Logarithm "Logarithm") of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution "Log-normal distribution"). Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different [random](https://en.wikipedia.org/wiki/Random "Random") factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called [Gibrat's law](https://en.wikipedia.org/wiki/Gibrat%27s_law "Gibrat's law").
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.[\[34\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Rempala-34)
## Beyond the classical framework
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=22 "Edit section: Beyond the classical framework")\]
Asymptotic normality, that is, [convergence](https://en.wikipedia.org/wiki/Convergence_in_distribution "Convergence in distribution") to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
### Convex body
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=23 "Edit section: Convex body")\]
**Theorem**âThere exists a sequence *Δn* â 0 for which the following holds. Let *n* â„ 1, and let random variables *X*1, ..., *Xn* have a [log-concave](https://en.wikipedia.org/wiki/Logarithmically_concave_function "Logarithmically concave function") [joint density](https://en.wikipedia.org/wiki/Joint_density_function "Joint density function") f such that *f*(*x*1, ..., *xn*) = *f*(\|*x*1\|, ..., \|*xn*\|) for all *x*1, ..., *xn*, and E(*X*2
*k*) = 1 for all *k* = 1, ..., *n*. Then the distribution of
X 1 \+ ⯠\+ X n n {\\displaystyle {\\frac {X\_{1}+\\cdots +X\_{n}}{\\sqrt {n}}}} 
is Δn\-close to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  in the [total variation distance](https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures "Total variation distance of probability measures").[\[35\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2007Theorem_1.2-35)
These two Δn\-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Another example: *f*(*x*1, ..., *xn*) = const · exp(â(\|*x*1\|*α* + ⯠+ \|*xn*\|*α*)*ÎČ*) where *α* \> 1 and *αÎČ* \> 1. If *ÎČ* = 1 then *f*(*x*1, ..., *xn*) factorizes into const · exp (â\|*x*1\|*α*) ⊠exp(â\|*xn*\|*α*), which means *X*1, ..., *Xn* are independent. In general, however, they are dependent.
The condition *f*(*x*1, ..., *xn*) = *f*(\|*x*1\|, ..., \|*xn*\|) ensures that *X*1, ..., *Xn* are of zero mean and [uncorrelated](https://en.wikipedia.org/wiki/Uncorrelated "Uncorrelated");\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] still, they need not be independent, nor even [pairwise independent](https://en.wikipedia.org/wiki/Pairwise_independence "Pairwise independence").\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] By the way, pairwise independence cannot replace independence in the classical central limit theorem.[\[36\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Section_2.4,_Example_4.5-36)
Here is a [BerryâEsseen](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem") type result.
**Theorem**âLet *X*1, ..., *Xn* satisfy the assumptions of the previous theorem, then[\[37\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2008Theorem_1-37)
\| P ( a †X 1 \+ ⯠\+ X n n †b ) â 1 2 Ï â« a b e â 1 2 t 2 d t \| †C n {\\displaystyle \\left\|\\mathbb {P} \\left(a\\leq {\\frac {X\_{1}+\\cdots +X\_{n}}{\\sqrt {n}}}\\leq b\\right)-{\\frac {1}{\\sqrt {2\\pi }}}\\int \_{a}^{b}e^{-{\\frac {1}{2}}t^{2}}\\,dt\\right\|\\leq {\\frac {C}{n}}} 
for all *a* \< *b*; here C is a [universal (absolute) constant](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant"). Moreover, for every *c*1, ..., *cn* â **R** such that *c*2
1 + ⯠+ *c*2
*n* = 1,
\| P ( a †c 1 X 1 \+ ⯠\+ c n X n †b ) â 1 2 Ï â« a b e â 1 2 t 2 d t \| †C ( c 1 4 \+ ⯠\+ c n 4 ) . {\\displaystyle \\left\|\\mathbb {P} \\left(a\\leq c\_{1}X\_{1}+\\cdots +c\_{n}X\_{n}\\leq b\\right)-{\\frac {1}{\\sqrt {2\\pi }}}\\int \_{a}^{b}e^{-{\\frac {1}{2}}t^{2}}\\,dt\\right\|\\leq C\\left(c\_{1}^{4}+\\dots +c\_{n}^{4}\\right).} 
The distribution of â *X*1 + ⯠+ *Xn*/â*n*â need not be approximately normal (in fact, it can be uniform).[\[38\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2007Theorem_1.1-38) However, the distribution of *c*1*X*1 + ⯠+ *cnXn* is close to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  (in the total variation distance) for most vectors (*c*1, ..., *cn*) according to the uniform distribution on the sphere *c*2
1 + ⯠+ *c*2
*n* = 1.
### Lacunary trigonometric series
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=24 "Edit section: Lacunary trigonometric series")\]
**Theorem ([Salem](https://en.wikipedia.org/wiki/Rapha%C3%ABl_Salem "RaphaĂ«l Salem")â[Zygmund](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund"))**âLet U be a random variable distributed uniformly on (0,2Ï), and *Xk* = *rk* cos(*nkU* + *ak*), where
- nk satisfy the lacunarity condition: there exists *q* \> 1 such that *n**k* + 1 â„ *qn**k* for all k,
- rk are such that
r
1
2
\+
r
2
2
\+
âŻ
\=
â
and
r
k
2
r
1
2
\+
âŻ
\+
r
k
2
â
0
,
{\\displaystyle r\_{1}^{2}+r\_{2}^{2}+\\cdots =\\infty \\quad {\\text{ and }}\\quad {\\frac {r\_{k}^{2}}{r\_{1}^{2}+\\cdots +r\_{k}^{2}}}\\to 0,}
- 0 †*a**k* \< 2Ï.
Then[\[39\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Zygmund-39)[\[40\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEGaposhkin1966Theorem_2.1.13-40)
X 1 \+ ⯠\+ X k r 1 2 \+ ⯠\+ r k 2 {\\displaystyle {\\frac {X\_{1}+\\cdots +X\_{k}}{\\sqrt {r\_{1}^{2}+\\cdots +r\_{k}^{2}}}}} 
converges in distribution to N ( 0 , 1 2 ) {\\textstyle {\\mathcal {N}}{\\big (}0,{\\frac {1}{2}}{\\big )}} .
### Gaussian polytopes
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=25 "Edit section: Gaussian polytopes")\]
**Theorem**âLet *A*1, ..., *A**n* be independent random points on the plane **R**2 each having the two-dimensional standard normal distribution. Let Kn be the [convex hull](https://en.wikipedia.org/wiki/Convex_hull "Convex hull") of these points, and Xn the area of Kn Then[\[41\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.1-41)
X n â E ⥠( X n ) Var ⥠( X n ) {\\displaystyle {\\frac {X\_{n}-\\operatorname {E} (X\_{n})}{\\sqrt {\\operatorname {Var} (X\_{n})}}}}  converges in distribution to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  as n tends to infinity.
The same also holds in all dimensions greater than 2.
The [polytope](https://en.wikipedia.org/wiki/Convex_polytope "Convex polytope") Kn is called a Gaussian [random polytope](https://en.wikipedia.org/wiki/Random_polytope "Random polytope").
A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.[\[42\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.2-42)
### Linear functions of orthogonal matrices
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=26 "Edit section: Linear functions of orthogonal matrices")\]
A linear function of a matrix **M** is a linear combination of its elements (with given coefficients), **M** ⊠tr(**AM**) where **A** is the matrix of the coefficients; see [Trace (linear algebra)\#Inner product](https://en.wikipedia.org/wiki/Trace_\(linear_algebra\)#Inner_product "Trace (linear algebra)").
A random [orthogonal matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix "Orthogonal matrix") is said to be distributed uniformly, if its distribution is the normalized [Haar measure](https://en.wikipedia.org/wiki/Haar_measure "Haar measure") on the [orthogonal group](https://en.wikipedia.org/wiki/Orthogonal_group "Orthogonal group") O(*n*,**R**); see [Rotation matrix\#Uniform random rotation matrices](https://en.wikipedia.org/wiki/Rotation_matrix#Uniform_random_rotation_matrices "Rotation matrix").
**Theorem**âLet **M** be a random orthogonal *n* Ă *n* matrix distributed uniformly, and **A** a fixed *n* Ă *n* matrix such that tr(**AA**\*) = *n*, and let *X* = tr(**AM**). Then[\[43\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Meckes-43) the distribution of X is close to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  in the total variation metric up to\[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")*\] â 2â3/*n* â 1â .
### Subsequences
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=27 "Edit section: Subsequences")\]
**Theorem**âLet random variables *X*1, *X*2, ... â *L*2(Ω) be such that *Xn* â 0 [weakly](https://en.wikipedia.org/wiki/Weak_convergence_\(Hilbert_space\) "Weak convergence (Hilbert space)") in *L*2(Ω) and *X*
*n* â 1 weakly in *L*1(Ω). Then there exist integers *n*1 \< *n*2 \< ⯠such that
X n 1 \+ ⯠\+ X n k k {\\displaystyle {\\frac {X\_{n\_{1}}+\\cdots +X\_{n\_{k}}}{\\sqrt {k}}}} 
converges in distribution to N ( 0 , 1 ) {\\textstyle {\\mathcal {N}}(0,1)}  as k tends to infinity.[\[44\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEGaposhkin1966Sect._1.5-44)
### Random walk on a crystal lattice
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=28 "Edit section: Random walk on a crystal lattice")\]
The central limit theorem may be established for the simple [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.[\[45\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-45)[\[46\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-46)
## Applications and examples
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=29 "Edit section: Applications and examples")\]
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample [statistics](https://en.wikipedia.org/wiki/Statistic "Statistic") to the normal distribution in controlled experiments.
[](https://en.wikipedia.org/wiki/File:Dice_sum_central_limit_theorem.svg)
Comparison of probability density functions *p*(*k*) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
[](https://en.wikipedia.org/wiki/File:Empirical_CLT_-_Figure_-_040711.jpg)
This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the [chi-squared](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test "Pearson's chi-squared test") values that quantify the goodness of the fit (the fit is good if the reduced [chi-squared](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test "Pearson's chi-squared test") value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/â*n*), which is called the standard deviation of the mean (since it refers to the spread of sample means).
[](https://en.wikipedia.org/wiki/File:Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg)
Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.
### Regression
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=30 "Edit section: Regression")\]
[Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis"), and in particular [ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares"), specifies that a [dependent variable](https://en.wikipedia.org/wiki/Dependent_variable "Dependent variable") depends according to some function upon one or more [independent variables](https://en.wikipedia.org/wiki/Independent_variable "Independent variable"), with an additive [error term](https://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics "Errors and residuals in statistics"). Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.
### Other illustrations
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=31 "Edit section: Other illustrations")\]
Main article: [Illustration of the central limit theorem](https://en.wikipedia.org/wiki/Illustration_of_the_central_limit_theorem "Illustration of the central limit theorem")
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.[\[47\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Marasinghe-47)
## History
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=32 "Edit section: History")\]
Dutch mathematician [Henk Tijms](https://en.wikipedia.org/wiki/Henk_Tijms "Henk Tijms") writes:[\[48\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Tijms-48)
> The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician [Abraham de Moivre](https://en.wikipedia.org/wiki/Abraham_de_Moivre "Abraham de Moivre") who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician [Pierre-Simon Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace") rescued it from obscurity in his monumental work *Théorie analytique des probabilités*, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.
Sir [Francis Galton](https://en.wikipedia.org/wiki/Francis_Galton "Francis Galton") described the Central Limit Theorem in this way:[\[49\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-49)
> I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by [George PĂłlya](https://en.wikipedia.org/wiki/George_P%C3%B3lya "George PĂłlya") in 1920 in the title of a paper.[\[50\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Polya1920-50)[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) PĂłlya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word *central* in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) The abstract of the paper *On the central limit theorem of calculus of probability and the problem of moments* by PĂłlya[\[50\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Polya1920-50) in 1920 translates as follows.
> The occurrence of the Gaussian probability density 1 = *e*â*x*2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by [Liapounoff](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov"). ...
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as [Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy "Augustin-Louis Cauchy")'s, [Bessel](https://en.wikipedia.org/wiki/Friedrich_Bessel "Friedrich Bessel")'s and [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "Siméon Denis Poisson")'s contributions, is provided by Hald.[\[52\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Hald-52) Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by [von Mises](https://en.wikipedia.org/wiki/Richard_von_Mises "Richard von Mises"), [Pólya](https://en.wikipedia.org/wiki/George_P%C3%B3lya "George Pólya"), [Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg"), [Lévy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul Lévy (mathematician)"), and [Cramér](https://en.wikipedia.org/wiki/Harald_Cram%C3%A9r "Harald Cramér") during the 1920s, are given by Hans Fischer.[\[53\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2-53) Le Cam describes a period around 1935.[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) Bernstein[\[54\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Bernstein-54) presents a historical discussion focusing on the work of [Pafnuty Chebyshev](https://en.wikipedia.org/wiki/Pafnuty_Chebyshev "Pafnuty Chebyshev") and his students [Andrey Markov](https://en.wikipedia.org/wiki/Andrey_Markov "Andrey Markov") and [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") that led to the first proofs of the CLT in a general setting.
A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of [Alan Turing](https://en.wikipedia.org/wiki/Alan_Turing "Alan Turing")'s 1934 Fellowship Dissertation for [King's College](https://en.wikipedia.org/wiki/King%27s_College,_Cambridge "King's College, Cambridge") at the [University of Cambridge](https://en.wikipedia.org/wiki/University_of_Cambridge "University of Cambridge"). Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.[\[55\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-55)
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=33 "Edit section: See also")\]
- [Asymptotic equipartition property](https://en.wikipedia.org/wiki/Asymptotic_equipartition_property "Asymptotic equipartition property")
- [Asymptotic distribution](https://en.wikipedia.org/wiki/Asymptotic_distribution "Asymptotic distribution")
- [Bates distribution](https://en.wikipedia.org/wiki/Bates_distribution "Bates distribution")
- [Benford's law](https://en.wikipedia.org/wiki/Benford%27s_law "Benford's law") â result of extension of CLT to product of random variables.
- [BerryâEsseen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem")
- [Central limit theorem for directional statistics](https://en.wikipedia.org/wiki/Central_limit_theorem_for_directional_statistics "Central limit theorem for directional statistics") â Central limit theorem applied to the case of directional statistics
- [Delta method](https://en.wikipedia.org/wiki/Delta_method "Delta method") â to compute the limit distribution of a function of a random variable.
- [ErdĆsâKac theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem "ErdĆsâKac theorem") â connects the number of prime factors of an integer with the normal probability distribution
- [FisherâTippettâGnedenko theorem](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem "FisherâTippettâGnedenko theorem") â limit theorem for extremum values (such as max{*Xn*})
- [IrwinâHall distribution](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution "IrwinâHall distribution")
- [Markov chain central limit theorem](https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem "Markov chain central limit theorem")
- [Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution")
- [Tweedie convergence theorem](https://en.wikipedia.org/wiki/Tweedie_distribution "Tweedie distribution") â a theorem that can be considered to bridge between the central limit theorem and the [Poisson convergence theorem](https://en.wikipedia.org/wiki/Poisson_convergence_theorem "Poisson convergence theorem")[\[56\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-J%C3%B8rgensen-1997-56)
- [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem")
## Notes
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=34 "Edit section: Notes")\]
1. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">[<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears. \(July_2023\)">page needed</span>]]</i>]</sup>_1-0)** [Fischer (2011)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFFischer2011), p. \[*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")*\].
2. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-2)**
Montgomery, Douglas C.; Runger, George C. (2014). *Applied Statistics and Probability for Engineers* (6th ed.). Wiley. p. 241. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781118539712](https://en.wikipedia.org/wiki/Special:BookSources/9781118539712 "Special:BookSources/9781118539712")
.
3. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-3)**
Rouaud, Mathieu (2013). [*Probability, Statistics and Estimation*](http://www.incertitudes.fr/book.pdf) (PDF). p. 10. [Archived](https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf) (PDF) from the original on 2022-10-09.
4. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995357_4-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), p. 357.
5. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBauer2001199Theorem_30.13_5-0)** [Bauer (2001)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBauer2001), p. 199, Theorem 30.13.
6. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995362_6-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), p. 362.
7. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-7)**
Robbins, Herbert (1948). ["The asymptotic distribution of the sum of a random number of random variables"](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-12/The-asymptotic-distribution-of-the-sum-of-a-random-number/bams/1183513324.full). *Bull. Amer. Math. Soc*. **54** (12): 1151â1161\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0002-9904-1948-09142-X](https://doi.org/10.1090%2FS0002-9904-1948-09142-X).
8. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-8)**
Chen, Louis H.Y.; Goldstein, Larry; Shao, Qi-Man (2011). *Normal Approximation by Stein's Method*. Berlin Heidelberg: Springer-Verlag. pp. 270â271\.
9. ^ [***a***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-vanderVaart_9-0) [***b***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-vanderVaart_9-1)
van der Vaart, A.W. (1998). *Asymptotic statistics*. New York, NY: Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-49603-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-49603-2 "Special:BookSources/978-0-521-49603-2")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [98015176](https://lccn.loc.gov/98015176).
10. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-10)**
[OâDonnell, Ryan](https://en.wikipedia.org/wiki/Ryan_O%27Donnell_\(computer_scientist\) "Ryan O'Donnell (computer scientist)") (2014). ["Theorem 5.38"](https://web.archive.org/web/20190408054104/http://www.contrib.andrew.cmu.edu/~ryanod/?p=866). Archived from [the original](http://www.contrib.andrew.cmu.edu/~ryanod/?p=866) on 2019-04-08. Retrieved 2017-10-18.
11. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-11)**
Bentkus, V. (2005). "A Lyapunov-type bound in
R
d
{\\displaystyle \\mathbb {R} ^{d}}
". *Theory Probab. Appl*. **49** (2): 311â323\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1137/S0040585X97981123](https://doi.org/10.1137%2FS0040585X97981123).
12. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-12)**
Le Cam, L. (February 1986). "The Central Limit Theorem around 1935". *Statistical Science*. **1** (1): 78â91\. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2245503](https://www.jstor.org/stable/2245503).
13. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-13)**
LĂ©vy, Paul (1937). *Theorie de l'addition des variables aleatoires* \[*Combination theory of unpredictable variables*\] (in French). Paris: Gauthier-Villars.
14. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-14)**
Gnedenko, Boris Vladimirovich; Kologorov, AndreÄ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968). *Limit distributions for sums of independent random variables*. Reading, MA: Addison-wesley.
15. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-15)**
Nolan, John P. (2020). [*Univariate stable distributions, Models for Heavy Tailed Data*](https://doi.org/10.1007/978-3-030-52915-4). Springer Series in Operations Research and Financial Engineering. Switzerland: Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-52915-4](https://doi.org/10.1007%2F978-3-030-52915-4). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-030-52914-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-52914-7 "Special:BookSources/978-3-030-52914-7")
. [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [226648987](https://api.semanticscholar.org/CorpusID:226648987).
16. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995Theorem_27.4_16-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), Theorem 27.4.
17. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Sect._7.7\(c\),_Theorem_7.8_17-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Sect. 7.7(c), Theorem 7.8.
18. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4_18-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Sect. 7.7, Theorem 7.4.
19. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995Theorem_35.12_19-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), Theorem 35.12.
20. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-20)**
Lemons, Don (2003). [*An Introduction to Stochastic Processes in Physics*](https://jhupbooks.press.jhu.edu/content/introduction-stochastic-processes-physics). Johns Hopkins University Press. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.56021/9780801868665](https://doi.org/10.56021%2F9780801868665). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780801876387](https://en.wikipedia.org/wiki/Special:BookSources/9780801876387 "Special:BookSources/9780801876387")
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21. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-stein1972_21-0)**
[Stein, C.](https://en.wikipedia.org/wiki/Charles_Stein_\(statistician\) "Charles Stein (statistician)") (1972). ["A bound for the error in the normal approximation to the distribution of a sum of dependent random variables"](https://projecteuclid.org/euclid.bsmsp/1200514239). *Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability*. **6** (2): 583â602\. [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0402873](https://mathscinet.ams.org/mathscinet-getitem?mr=0402873). [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0278\.60026](https://zbmath.org/?format=complete&q=an:0278.60026).
22. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-22)**
Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). *Normal approximation by Stein's method*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-642-15006-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-15006-7 "Special:BookSources/978-3-642-15006-7")
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23. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-ABBN_23-0)**
[Artstein, S.](https://en.wikipedia.org/wiki/Shiri_Artstein "Shiri Artstein"); [Ball, K.](https://en.wikipedia.org/wiki/Keith_Martin_Ball "Keith Martin Ball"); [Barthe, F.](https://en.wikipedia.org/wiki/Franck_Barthe "Franck Barthe"); [Naor, A.](https://en.wikipedia.org/wiki/Assaf_Naor "Assaf Naor") (2004). ["Solution of Shannon's Problem on the Monotonicity of Entropy"](https://doi.org/10.1090%2FS0894-0347-04-00459-X). *Journal of the American Mathematical Society*. **17** (4): 975â982\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0894-0347-04-00459-X](https://doi.org/10.1090%2FS0894-0347-04-00459-X).
24. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-24)**
Brewer, J. K. (1985). "Behavioral statistics textbooks: Source of myths and misconceptions?". *Journal of Educational Statistics*. **10** (3): 252â268\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3102/10769986010003252](https://doi.org/10.3102%2F10769986010003252). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119611584](https://api.semanticscholar.org/CorpusID:119611584).
25. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-25)** Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, *American Educational Research Association* lecture 19 April 1995
26. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-26)**
Sotos, A. E. C.; Vanhoof, S.; Van den Noortgate, W.; Onghena, P. (2007). ["Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education"](https://lirias.kuleuven.be/handle/123456789/136347). *Educational Research Review*. **2** (2): 98â113\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.edurev.2007.04.001](https://doi.org/10.1016%2Fj.edurev.2007.04.001).
27. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-27)**
["Sampling distribution of the sample mean"](https://web.archive.org/web/20230602200310/https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean). *Khan Academy*. 2 June 2023. Archived from [the original](https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean) (video) on 2023-06-02. Retrieved 2023-10-08.
28. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-28)**
Rosenthal, Jeffrey Seth (2000). *A First Look at Rigorous Probability Theory*. World Scientific. Theorem 5.3.4, p. 47. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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29. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-29)**
Johnson, Oliver Thomas (2004). *Information Theory and the Central Limit Theorem*. Imperial College Press. p. 88. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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30. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Uchaikin_30-0)**
Uchaikin, Vladimir V.; Zolotarev, V.M. (1999). *Chance and Stability: Stable distributions and their applications*. VSP. pp. 61â62\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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31. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-31)**
Borodin, A. N.; Ibragimov, I. A.; Sudakov, V. N. (1995). *Limit Theorems for Functionals of Random Walks*. AMS Bookstore. Theorem 1.1, p. 8. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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32. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-32)**
Petrov, V. V. (1976). [*Sums of Independent Random Variables*](https://books.google.com/books?id=zSDqCAAAQBAJ). New York-Heidelberg: Springer-Verlag. ch. 7. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9783642658099](https://en.wikipedia.org/wiki/Special:BookSources/9783642658099 "Special:BookSources/9783642658099")
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33. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-33)**
Hew, Patrick Chisan (2017). "Asymptotic distribution of rewards accumulated by alternating renewal processes". *Statistics and Probability Letters*. **129**: 355â359\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.spl.2017.06.027](https://doi.org/10.1016%2Fj.spl.2017.06.027).
34. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Rempala_34-0)**
Rempala, G.; Wesolowski, J. (2002). ["Asymptotics of products of sums and *U*\-statistics"](https://projecteuclid.org/journals/electronic-communications-in-probability/volume-7/issue-none/Asymptotics-for-Products-of-Sums-and-U-statistics/10.1214/ECP.v7-1046.pdf) (PDF). *Electronic Communications in Probability*. **7**: 47â54\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1214/ecp.v7-1046](https://doi.org/10.1214%2Fecp.v7-1046).
35. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2007Theorem_1.2_35-0)** [Klartag (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2007), Theorem 1.2.
36. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Section_2.4,_Example_4.5_36-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Section 2.4, Example 4.5.
37. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2008Theorem_1_37-0)** [Klartag (2008)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2008), Theorem 1.
38. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2007Theorem_1.1_38-0)** [Klartag (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2007), Theorem 1.1.
39. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Zygmund_39-0)**
[Zygmund, Antoni](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund") (2003) \[1959\]. [*Trigonometric Series*](https://en.wikipedia.org/wiki/Trigonometric_Series "Trigonometric Series"). Cambridge University Press. vol. II, sect. XVI.5, Theorem 5-5. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-521-89053-5](https://en.wikipedia.org/wiki/Special:BookSources/0-521-89053-5 "Special:BookSources/0-521-89053-5")
.
40. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEGaposhkin1966Theorem_2.1.13_40-0)** [Gaposhkin (1966)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFGaposhkin1966), Theorem 2.1.13.
41. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.1_41-0)** [BĂĄrĂĄny & Vu (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFB%C3%A1r%C3%A1nyVu2007), Theorem 1.1.
42. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.2_42-0)** [BĂĄrĂĄny & Vu (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFB%C3%A1r%C3%A1nyVu2007), Theorem 1.2.
43. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Meckes_43-0)**
[Meckes, Elizabeth](https://en.wikipedia.org/wiki/Elizabeth_Meckes "Elizabeth Meckes") (2008). "Linear functions on the classical matrix groups". *Transactions of the American Mathematical Society*. **360** (10): 5355â5366\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0509441](https://arxiv.org/abs/math/0509441). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0002-9947-08-04444-9](https://doi.org/10.1090%2FS0002-9947-08-04444-9). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [11981408](https://api.semanticscholar.org/CorpusID:11981408).
44. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEGaposhkin1966Sect._1.5_44-0)** [Gaposhkin (1966)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFGaposhkin1966), Sect. 1.5.
45. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-45)**
Kotani, M.; [Sunada, Toshikazu](https://en.wikipedia.org/wiki/Toshikazu_Sunada "Toshikazu Sunada") (2003). *Spectral geometry of crystal lattices*. Vol. 338. Contemporary Math. pp. 271â305\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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46. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-46)**
[Sunada, Toshikazu](https://en.wikipedia.org/wiki/Toshikazu_Sunada "Toshikazu Sunada") (2012). *Topological Crystallography â With a View Towards Discrete Geometric Analysis*. Surveys and Tutorials in the Applied Mathematical Sciences. Vol. 6. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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47. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Marasinghe_47-0)**
Marasinghe, M.; Meeker, W.; Cook, D.; Shin, T. S. (August 1994). *Using graphics and simulation to teach statistical concepts*. Annual meeting of the American Statistician Association, Toronto, Canada.
48. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Tijms_48-0)**
Henk, Tijms (2004). *Understanding Probability: Chance Rules in Everyday Life*. Cambridge: Cambridge University Press. p. 169. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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49. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-49)**
Galton, F. (1889). [*Natural Inheritance*](https://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm). p. 66.
50. ^ [***a***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Polya1920_50-0) [***b***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Polya1920_50-1)
[PĂłlya, George](https://en.wikipedia.org/wiki/George_P%C3%B3lya "George PĂłlya") (1920). ["Ăber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem"](https://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0008) \[On the central limit theorem of probability calculation and the problem of moments\]. *[Mathematische Zeitschrift](https://en.wikipedia.org/wiki/Mathematische_Zeitschrift "Mathematische Zeitschrift")* (in German). **8** (3â4\): 171â181\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01206525](https://doi.org/10.1007%2FBF01206525). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [123063388](https://api.semanticscholar.org/CorpusID:123063388).
51. ^ [***a***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-LC1986_51-0) [***b***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-LC1986_51-1) [***c***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-LC1986_51-2)
[Le Cam, Lucien](https://en.wikipedia.org/wiki/Lucien_Le_Cam "Lucien Le Cam") (1986). ["The central limit theorem around 1935"](http://projecteuclid.org/euclid.ss/1177013818). *Statistical Science*. **1** (1): 78â91\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1214/ss/1177013818](https://doi.org/10.1214%2Fss%2F1177013818).
52. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Hald_52-0)**
Hald, Andreas (22 April 1998). [*A History of Mathematical Statistics from 1750 to 1930*](http://www.gbv.de/dms/goettingen/229762905.pdf) (PDF). Wiley. chapter 17. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0471179122](https://en.wikipedia.org/wiki/Special:BookSources/978-0471179122 "Special:BookSources/978-0471179122")
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53. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2_53-0)** [Fischer (2011)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFFischer2011), Chapter 2; Chapter 5.2.
54. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Bernstein_54-0)**
[Bernstein, S. N.](https://en.wikipedia.org/wiki/Sergei_Natanovich_Bernstein "Sergei Natanovich Bernstein") (1945). "On the work of P. L. Chebyshev in Probability Theory". In Bernstein., S. N. (ed.). *Nauchnoe Nasledie P. L. Chebysheva. Vypusk Pervyi: Matematika* \[*The Scientific Legacy of P. L. Chebyshev. Part I: Mathematics*\] (in Russian). Moscow & Leningrad: Academiya Nauk SSSR. p. 174.
55. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-55)**
Zabell, S. L. (1995). "Alan Turing and the Central Limit Theorem". *American Mathematical Monthly*. **102** (6): 483â494\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/00029890.1995.12004608](https://doi.org/10.1080%2F00029890.1995.12004608).
56. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-J%C3%B8rgensen-1997_56-0)**
JĂžrgensen, Bent (1997). *The Theory of Dispersion Models*. Chapman & Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0412997112](https://en.wikipedia.org/wiki/Special:BookSources/978-0412997112 "Special:BookSources/978-0412997112")
.
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=35 "Edit section: References")\]
- [BĂĄrĂĄny, Imre](https://en.wikipedia.org/wiki/Imre_B%C3%A1r%C3%A1ny "Imre BĂĄrĂĄny"); Vu, Van (2007). "Central limit theorems for Gaussian polytopes". *Annals of Probability*. **35** (4). Institute of Mathematical Statistics: 1593â1621\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0610192](https://arxiv.org/abs/math/0610192). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1214/009117906000000791](https://doi.org/10.1214%2F009117906000000791). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [9128253](https://api.semanticscholar.org/CorpusID:9128253).
- Bauer, Heinz (2001). *Measure and Integration Theory*. Berlin: de Gruyter. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[3110167190](https://en.wikipedia.org/wiki/Special:BookSources/3110167190 "Special:BookSources/3110167190")
.
- Billingsley, Patrick (1995). *Probability and Measure* (3rd ed.). John Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-471-00710-2](https://en.wikipedia.org/wiki/Special:BookSources/0-471-00710-2 "Special:BookSources/0-471-00710-2")
.
- Bradley, Richard (2005). "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions". *Probability Surveys*. **2**: 107â144\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0511078](https://arxiv.org/abs/math/0511078). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2005math.....11078B](https://ui.adsabs.harvard.edu/abs/2005math.....11078B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1214/154957805100000104](https://doi.org/10.1214%2F154957805100000104). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [8395267](https://api.semanticscholar.org/CorpusID:8395267).
- Bradley, Richard (2007). *Introduction to Strong Mixing Conditions* (1st ed.). Heber City, UT: Kendrick Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-9740427-9-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9740427-9-4 "Special:BookSources/978-0-9740427-9-4")
.
- Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008). ["Central Limit Theorem: New SOCR Applet and Demonstration Activity"](https://web.archive.org/web/20160303185802/http://www.amstat.org/publications/jse/v16n2/dinov.html). *Journal of Statistics Education*. **16** (2). ASA: 1â15\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/10691898.2008.11889560](https://doi.org/10.1080%2F10691898.2008.11889560). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3152447](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3152447). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [21833159](https://pubmed.ncbi.nlm.nih.gov/21833159). Archived from [the original](http://www.amstat.org/publications/jse/v16n2/dinov.html) on 2016-03-03. Retrieved 2008-08-23.
- [Durrett, Richard](https://en.wikipedia.org/wiki/Rick_Durrett "Rick Durrett") (2004). *Probability: theory and examples* (3rd ed.). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0521765390](https://en.wikipedia.org/wiki/Special:BookSources/0521765390 "Special:BookSources/0521765390")
.
- Fischer, Hans (2011). [*A History of the Central Limit Theorem: From Classical to Modern Probability Theory*](http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf) (PDF). Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-0-387-87857-7](https://doi.org/10.1007%2F978-0-387-87857-7). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-87856-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-87856-0 "Special:BookSources/978-0-387-87856-0")
. [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [2743162](https://mathscinet.ams.org/mathscinet-getitem?mr=2743162). [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [1226\.60004](https://zbmath.org/?format=complete&q=an:1226.60004). [Archived](https://web.archive.org/web/20171031171033/http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf) (PDF) from the original on 2017-10-31.
- Gaposhkin, V. F. (1966). "Lacunary series and independent functions". *Russian Mathematical Surveys*. **21** (6): 1â82\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1966RuMaS..21....1G](https://ui.adsabs.harvard.edu/abs/1966RuMaS..21....1G). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1070/RM1966v021n06ABEH001196](https://doi.org/10.1070%2FRM1966v021n06ABEH001196). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [250833638](https://api.semanticscholar.org/CorpusID:250833638).
.
- Klartag, Bo'az (2007). "A central limit theorem for convex sets". *Inventiones Mathematicae*. **168** (1): 91â131\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0605014](https://arxiv.org/abs/math/0605014). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2007InMat.168...91K](https://ui.adsabs.harvard.edu/abs/2007InMat.168...91K). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s00222-006-0028-8](https://doi.org/10.1007%2Fs00222-006-0028-8). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119169773](https://api.semanticscholar.org/CorpusID:119169773).
- Klartag, Bo'az (2008). "A BerryâEsseen type inequality for convex bodies with an unconditional basis". *Probability Theory and Related Fields*. **145** (1â2\): 1â33\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[0705\.0832](https://arxiv.org/abs/0705.0832). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s00440-008-0158-6](https://doi.org/10.1007%2Fs00440-008-0158-6). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [10163322](https://api.semanticscholar.org/CorpusID:10163322).
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=36 "Edit section: External links")\]
[](https://en.wikipedia.org/wiki/File:Commons-logo.svg)
Wikimedia Commons has media related to [Central limit theorem](https://commons.wikimedia.org/wiki/Category:Central_limit_theorem "commons:Category:Central limit theorem").
- [Central Limit Theorem](https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/central-limit-theorem) at Khan Academy
- ["Central limit theorem"](https://www.encyclopediaofmath.org/index.php?title=Central_limit_theorem). *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*. [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"). 2001 \[1994\].
- [Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Central Limit Theorem"](https://mathworld.wolfram.com/CentralLimitTheorem.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
- [A music video demonstrating the central limit theorem with a Galton board](https://www.mctague.org/carl/blog/2021/04/23/central-limit-theorem/) by Carl McTague
| [v](https://en.wikipedia.org/wiki/Template:Statistics "Template:Statistics") [t](https://en.wikipedia.org/wiki/Template_talk:Statistics "Template talk:Statistics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Statistics "Special:EditPage/Template:Statistics")[Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") | |
|---|---|
| [Outline](https://en.wikipedia.org/wiki/Outline_of_statistics "Outline of statistics") [Index](https://en.wikipedia.org/wiki/List_of_statistics_articles "List of statistics articles") | |
| [Descriptive statistics](https://en.wikipedia.org/wiki/Descriptive_statistics "Descriptive statistics") | |
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| [Continuous data](https://en.wikipedia.org/wiki/Continuous_probability_distribution "Continuous probability distribution") | |
| | |
| [Center](https://en.wikipedia.org/wiki/Central_tendency "Central tendency") | [Mean](https://en.wikipedia.org/wiki/Mean "Mean") [Arithmetic](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean") [Arithmetic-Geometric](https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean "Arithmeticâgeometric mean") [Contraharmonic](https://en.wikipedia.org/wiki/Contraharmonic_mean "Contraharmonic mean") [Cubic](https://en.wikipedia.org/wiki/Cubic_mean "Cubic mean") [Generalized/power](https://en.wikipedia.org/wiki/Generalized_mean "Generalized mean") [Geometric](https://en.wikipedia.org/wiki/Geometric_mean "Geometric mean") [Harmonic](https://en.wikipedia.org/wiki/Harmonic_mean "Harmonic mean") [Heronian](https://en.wikipedia.org/wiki/Heronian_mean "Heronian mean") [Heinz](https://en.wikipedia.org/wiki/Heinz_mean "Heinz mean") [Lehmer](https://en.wikipedia.org/wiki/Lehmer_mean "Lehmer mean") [Median](https://en.wikipedia.org/wiki/Median "Median") [Mode](https://en.wikipedia.org/wiki/Mode_\(statistics\) "Mode (statistics)") |
| [Dispersion](https://en.wikipedia.org/wiki/Statistical_dispersion "Statistical dispersion") | [Average absolute deviation](https://en.wikipedia.org/wiki/Average_absolute_deviation "Average absolute deviation") [Coefficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation "Coefficient of variation") [Interquartile range](https://en.wikipedia.org/wiki/Interquartile_range "Interquartile range") [Percentile](https://en.wikipedia.org/wiki/Percentile "Percentile") [Range](https://en.wikipedia.org/wiki/Range_\(statistics\) "Range (statistics)") [Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") [Variance](https://en.wikipedia.org/wiki/Variance#Sample_variance "Variance") |
| [Shape](https://en.wikipedia.org/wiki/Shape_of_the_distribution "Shape of the distribution") | [Central limit theorem]() [Moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") [Kurtosis](https://en.wikipedia.org/wiki/Kurtosis "Kurtosis") [L-moments](https://en.wikipedia.org/wiki/L-moment "L-moment") [Skewness](https://en.wikipedia.org/wiki/Skewness "Skewness") |
| [Count data](https://en.wikipedia.org/wiki/Count_data "Count data") | [Index of dispersion](https://en.wikipedia.org/wiki/Index_of_dispersion "Index of dispersion") |
| Summary tables | [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Frequency distribution](https://en.wikipedia.org/wiki/Frequency_distribution "Frequency distribution") [Grouped data](https://en.wikipedia.org/wiki/Grouped_data "Grouped data") |
| [Dependence](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Pearson product-moment correlation](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient "Pearson correlation coefficient") [Rank correlation](https://en.wikipedia.org/wiki/Rank_correlation "Rank correlation") [Kendall's Ï](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient "Kendall rank correlation coefficient") [Spearman's Ï](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient "Spearman's rank correlation coefficient") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") |
| [Graphics](https://en.wikipedia.org/wiki/Statistical_graphics "Statistical graphics") | [Bar chart](https://en.wikipedia.org/wiki/Bar_chart "Bar chart") [Biplot](https://en.wikipedia.org/wiki/Biplot "Biplot") [Box plot](https://en.wikipedia.org/wiki/Box_plot "Box plot") [Control chart](https://en.wikipedia.org/wiki/Control_chart "Control chart") [Correlogram](https://en.wikipedia.org/wiki/Correlogram "Correlogram") [Fan chart](https://en.wikipedia.org/wiki/Fan_chart_\(statistics\) "Fan chart (statistics)") [Forest plot](https://en.wikipedia.org/wiki/Forest_plot "Forest plot") [Histogram](https://en.wikipedia.org/wiki/Histogram "Histogram") [Pie chart](https://en.wikipedia.org/wiki/Pie_chart "Pie chart") [QâQ plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot "QâQ plot") [Radar chart](https://en.wikipedia.org/wiki/Radar_chart "Radar chart") [Run chart](https://en.wikipedia.org/wiki/Run_chart "Run chart") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") [Stem-and-leaf display](https://en.wikipedia.org/wiki/Stem-and-leaf_display "Stem-and-leaf display") [Violin plot](https://en.wikipedia.org/wiki/Violin_plot "Violin plot") [Heatmap](https://en.wikipedia.org/wiki/Heatmap "Heatmap") [Scatter Plot Matrix](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") [ECDF plot](https://en.wikipedia.org/wiki/Empirical_distribution_function "Empirical distribution function") |
| [Statistical data processing](https://en.wikipedia.org/wiki/Data_preprocessing "Data preprocessing") | |
| | |
| [Transformations](https://en.wikipedia.org/wiki/Data_transformation_\(statistics\) "Data transformation (statistics)") | [Data transformation](https://en.wikipedia.org/wiki/Data_transformation_\(statistics\) "Data transformation (statistics)") [Log transformation](https://en.wikipedia.org/w/index.php?title=Log_transformation&action=edit&redlink=1 "Log transformation (page does not exist)") [Power transform](https://en.wikipedia.org/wiki/Power_transform "Power transform") [BoxâCox transformation](https://en.wikipedia.org/wiki/Box%E2%80%93Cox_transformation "BoxâCox transformation") [YeoâJohnson transformation](https://en.wikipedia.org/wiki/Yeo%E2%80%93Johnson_transformation "YeoâJohnson transformation") [Variance-stabilizing transformation](https://en.wikipedia.org/wiki/Variance-stabilizing_transformation "Variance-stabilizing transformation") [Anscombe transform](https://en.wikipedia.org/wiki/Anscombe_transform "Anscombe transform") [Fisher transformation](https://en.wikipedia.org/wiki/Fisher_transformation "Fisher transformation") |
| [Scaling and normalization](https://en.wikipedia.org/wiki/Feature_scaling "Feature scaling") | [Feature scaling](https://en.wikipedia.org/wiki/Feature_scaling "Feature scaling") [Normalization](https://en.wikipedia.org/wiki/Normalization_\(statistics\) "Normalization (statistics)") [Standardization (z-score)](https://en.wikipedia.org/wiki/Standard_score "Standard score") [Minâmax normalization](https://en.wikipedia.org/w/index.php?title=Min%E2%80%93max_normalization&action=edit&redlink=1 "Minâmax normalization (page does not exist)") [Unit vector normalization](https://en.wikipedia.org/w/index.php?title=Unit_vector_normalization&action=edit&redlink=1 "Unit vector normalization (page does not exist)") |
| Data cleaning | [Data cleaning](https://en.wikipedia.org/wiki/Data_cleaning "Data cleaning") [Outlier](https://en.wikipedia.org/wiki/Outlier "Outlier") [Winsorizing](https://en.wikipedia.org/wiki/Winsorizing "Winsorizing") [Truncation](https://en.wikipedia.org/wiki/Truncation_\(statistics\) "Truncation (statistics)") [Missing data](https://en.wikipedia.org/wiki/Missing_data "Missing data") |
| Data reduction | [Dimensionality reduction](https://en.wikipedia.org/wiki/Dimensionality_reduction "Dimensionality reduction") [Principal component analysis](https://en.wikipedia.org/wiki/Principal_component_analysis "Principal component analysis") [Factor analysis](https://en.wikipedia.org/wiki/Factor_analysis "Factor analysis") |
| Time-series preprocessing | [Differencing](https://en.wikipedia.org/wiki/Differencing "Differencing") [Detrending](https://en.wikipedia.org/wiki/Detrending "Detrending") [Seasonal adjustment](https://en.wikipedia.org/wiki/Seasonal_adjustment "Seasonal adjustment") [Stationarity transformation](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") |
| [Data collection](https://en.wikipedia.org/wiki/Data_collection "Data collection") | |
| | |
| [Study design](https://en.wikipedia.org/wiki/Design_of_experiments "Design of experiments") | [Effect size](https://en.wikipedia.org/wiki/Effect_size "Effect size") [Missing data](https://en.wikipedia.org/wiki/Missing_data "Missing data") [Optimal design](https://en.wikipedia.org/wiki/Optimal_design "Optimal design") [Population](https://en.wikipedia.org/wiki/Statistical_population "Statistical population") [Replication](https://en.wikipedia.org/wiki/Replication_\(statistics\) "Replication (statistics)") [Sample size determination](https://en.wikipedia.org/wiki/Sample_size_determination "Sample size determination") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Statistical power](https://en.wikipedia.org/wiki/Statistical_power "Statistical power") |
| [Survey methodology](https://en.wikipedia.org/wiki/Survey_methodology "Survey methodology") | [Sampling](https://en.wikipedia.org/wiki/Sampling_\(statistics\) "Sampling (statistics)") [Cluster](https://en.wikipedia.org/wiki/Cluster_sampling "Cluster sampling") [Stratified](https://en.wikipedia.org/wiki/Stratified_sampling "Stratified sampling") [Opinion poll](https://en.wikipedia.org/wiki/Opinion_poll "Opinion poll") [Questionnaire](https://en.wikipedia.org/wiki/Questionnaire "Questionnaire") [Standard error](https://en.wikipedia.org/wiki/Standard_error "Standard error") |
| [Controlled experiments](https://en.wikipedia.org/wiki/Experiment "Experiment") | [Blocking](https://en.wikipedia.org/wiki/Blocking_\(statistics\) "Blocking (statistics)") [Factorial experiment](https://en.wikipedia.org/wiki/Factorial_experiment "Factorial experiment") [Interaction](https://en.wikipedia.org/wiki/Interaction_\(statistics\) "Interaction (statistics)") [Random assignment](https://en.wikipedia.org/wiki/Random_assignment "Random assignment") [Randomized controlled trial](https://en.wikipedia.org/wiki/Randomized_controlled_trial "Randomized controlled trial") [Randomized experiment](https://en.wikipedia.org/wiki/Randomized_experiment "Randomized experiment") [Scientific control](https://en.wikipedia.org/wiki/Scientific_control "Scientific control") |
| Adaptive designs | [Adaptive clinical trial](https://en.wikipedia.org/wiki/Adaptive_clinical_trial "Adaptive clinical trial") [Stochastic approximation](https://en.wikipedia.org/wiki/Stochastic_approximation "Stochastic approximation") [Up-and-down designs](https://en.wikipedia.org/wiki/Up-and-Down_Designs "Up-and-Down Designs") |
| [Observational studies](https://en.wikipedia.org/wiki/Observational_study "Observational study") | [Cohort study](https://en.wikipedia.org/wiki/Cohort_study "Cohort study") [Cross-sectional study](https://en.wikipedia.org/wiki/Cross-sectional_study "Cross-sectional study") [Natural experiment](https://en.wikipedia.org/wiki/Natural_experiment "Natural experiment") [Quasi-experiment](https://en.wikipedia.org/wiki/Quasi-experiment "Quasi-experiment") |
| [Statistical inference](https://en.wikipedia.org/wiki/Statistical_inference "Statistical inference") | |
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| [Statistical theory](https://en.wikipedia.org/wiki/Statistical_theory "Statistical theory") | [Population](https://en.wikipedia.org/wiki/Population_\(statistics\) "Population (statistics)") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") [Sampling distribution](https://en.wikipedia.org/wiki/Sampling_distribution "Sampling distribution") [Order statistic](https://en.wikipedia.org/wiki/Order_statistic "Order statistic") [Empirical distribution](https://en.wikipedia.org/wiki/Empirical_distribution_function "Empirical distribution function") [Density estimation](https://en.wikipedia.org/wiki/Density_estimation "Density estimation") [Statistical model](https://en.wikipedia.org/wiki/Statistical_model "Statistical model") [Model specification](https://en.wikipedia.org/wiki/Model_specification "Model specification") [L*p* space](https://en.wikipedia.org/wiki/Lp_space "Lp space") [Parameter](https://en.wikipedia.org/wiki/Statistical_parameter "Statistical parameter") [location](https://en.wikipedia.org/wiki/Location_parameter "Location parameter") [scale](https://en.wikipedia.org/wiki/Scale_parameter "Scale parameter") [shape](https://en.wikipedia.org/wiki/Shape_parameter "Shape parameter") [Parametric family](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") [Likelihood](https://en.wikipedia.org/wiki/Likelihood_function "Likelihood function") [(monotone)](https://en.wikipedia.org/wiki/Monotone_likelihood_ratio "Monotone likelihood ratio") [Locationâscale family](https://en.wikipedia.org/wiki/Location%E2%80%93scale_family "Locationâscale family") [Exponential family](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Completeness](https://en.wikipedia.org/wiki/Completeness_\(statistics\) "Completeness (statistics)") [Sufficiency](https://en.wikipedia.org/wiki/Sufficient_statistic "Sufficient statistic") [Statistical functional](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [U](https://en.wikipedia.org/wiki/U-statistic "U-statistic") [V](https://en.wikipedia.org/wiki/V-statistic "V-statistic") [Optimal decision](https://en.wikipedia.org/wiki/Optimal_decision "Optimal decision") [loss function](https://en.wikipedia.org/wiki/Loss_function "Loss function") [Efficiency](https://en.wikipedia.org/wiki/Efficiency_\(statistics\) "Efficiency (statistics)") [Statistical distance](https://en.wikipedia.org/wiki/Statistical_distance "Statistical distance") [divergence](https://en.wikipedia.org/wiki/Divergence_\(statistics\) "Divergence (statistics)") [Asymptotics](https://en.wikipedia.org/wiki/Asymptotic_theory_\(statistics\) "Asymptotic theory (statistics)") [Robustness](https://en.wikipedia.org/wiki/Robust_statistics "Robust statistics") |
| [Frequentist inference](https://en.wikipedia.org/wiki/Frequentist_inference "Frequentist inference") | |
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| [Point estimation](https://en.wikipedia.org/wiki/Point_estimation "Point estimation") | [Estimating equations](https://en.wikipedia.org/wiki/Estimating_equations "Estimating equations") [Maximum likelihood](https://en.wikipedia.org/wiki/Maximum_likelihood "Maximum likelihood") [Method of moments](https://en.wikipedia.org/wiki/Method_of_moments_\(statistics\) "Method of moments (statistics)") [M-estimator](https://en.wikipedia.org/wiki/M-estimator "M-estimator") [Minimum distance](https://en.wikipedia.org/wiki/Minimum_distance_estimation "Minimum distance estimation") [Unbiased estimators](https://en.wikipedia.org/wiki/Bias_of_an_estimator "Bias of an estimator") [Mean-unbiased minimum-variance](https://en.wikipedia.org/wiki/Minimum-variance_unbiased_estimator "Minimum-variance unbiased estimator") [RaoâBlackwellization](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem "RaoâBlackwell theorem") [LehmannâScheffĂ© theorem](https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem "LehmannâScheffĂ© theorem") [Median unbiased](https://en.wikipedia.org/wiki/Median-unbiased_estimator "Median-unbiased estimator") [Plug-in](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") |
| [Interval estimation](https://en.wikipedia.org/wiki/Interval_estimation "Interval estimation") | [Confidence interval](https://en.wikipedia.org/wiki/Confidence_interval "Confidence interval") [Pivot](https://en.wikipedia.org/wiki/Pivotal_quantity "Pivotal quantity") [Likelihood interval](https://en.wikipedia.org/wiki/Likelihood_interval "Likelihood interval") [Prediction interval](https://en.wikipedia.org/wiki/Prediction_interval "Prediction interval") [Tolerance interval](https://en.wikipedia.org/wiki/Tolerance_interval "Tolerance interval") [Resampling](https://en.wikipedia.org/wiki/Resampling_\(statistics\) "Resampling (statistics)") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [Jackknife](https://en.wikipedia.org/wiki/Jackknife_resampling "Jackknife resampling") |
| [Testing hypotheses](https://en.wikipedia.org/wiki/Statistical_hypothesis_testing "Statistical hypothesis testing") | [1- & 2-tails](https://en.wikipedia.org/wiki/One-_and_two-tailed_tests "One- and two-tailed tests") [Power](https://en.wikipedia.org/wiki/Power_\(statistics\) "Power (statistics)") [Uniformly most powerful test](https://en.wikipedia.org/wiki/Uniformly_most_powerful_test "Uniformly most powerful test") [Permutation test](https://en.wikipedia.org/wiki/Permutation_test "Permutation test") [Randomization test](https://en.wikipedia.org/wiki/Randomization_test "Randomization test") [Multiple comparisons](https://en.wikipedia.org/wiki/Multiple_comparisons "Multiple comparisons") |
| [Parametric tests](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") | [Likelihood-ratio](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Score/Lagrange multiplier](https://en.wikipedia.org/wiki/Score_test "Score test") [Wald](https://en.wikipedia.org/wiki/Wald_test "Wald test") |
| [Specific tests](https://en.wikipedia.org/wiki/List_of_statistical_tests "List of statistical tests") | |
| | |
| [*Z*\-test (normal)](https://en.wikipedia.org/wiki/Z-test "Z-test") [Student's *t*\-test](https://en.wikipedia.org/wiki/Student%27s_t-test "Student's t-test") [*F*\-test](https://en.wikipedia.org/wiki/F-test "F-test") | |
| [Goodness of fit](https://en.wikipedia.org/wiki/Goodness_of_fit "Goodness of fit") | [Chi-squared](https://en.wikipedia.org/wiki/Chi-squared_test "Chi-squared test") [*G*\-test](https://en.wikipedia.org/wiki/G-test "G-test") [KolmogorovâSmirnov](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test "KolmogorovâSmirnov test") [AndersonâDarling](https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test "AndersonâDarling test") [Lilliefors](https://en.wikipedia.org/wiki/Lilliefors_test "Lilliefors test") [JarqueâBera](https://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test "JarqueâBera test") [Normality (ShapiroâWilk)](https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test "ShapiroâWilk test") [Likelihood-ratio test](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Model selection](https://en.wikipedia.org/wiki/Model_selection "Model selection") [Cross validation](https://en.wikipedia.org/wiki/Cross-validation_\(statistics\) "Cross-validation (statistics)") [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion "Akaike information criterion") [BIC](https://en.wikipedia.org/wiki/Bayesian_information_criterion "Bayesian information criterion") |
| [Rank statistics](https://en.wikipedia.org/wiki/Rank_statistics "Rank statistics") | [Sign](https://en.wikipedia.org/wiki/Sign_test "Sign test") [Sample median](https://en.wikipedia.org/wiki/Sample_median "Sample median") [Signed rank (Wilcoxon)](https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test "Wilcoxon signed-rank test") [HodgesâLehmann estimator](https://en.wikipedia.org/wiki/Hodges%E2%80%93Lehmann_estimator "HodgesâLehmann estimator") [Rank sum (MannâWhitney)](https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test "MannâWhitney U test") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_statistics "Nonparametric statistics") [anova](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [1-way (KruskalâWallis)](https://en.wikipedia.org/wiki/Kruskal%E2%80%93Wallis_test "KruskalâWallis test") [2-way (Friedman)](https://en.wikipedia.org/wiki/Friedman_test "Friedman test") [Ordered alternative (JonckheereâTerpstra)](https://en.wikipedia.org/wiki/Jonckheere%27s_trend_test "Jonckheere's trend test") [Van der Waerden test](https://en.wikipedia.org/wiki/Van_der_Waerden_test "Van der Waerden test") |
| [Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference "Bayesian inference") | [Bayesian probability](https://en.wikipedia.org/wiki/Bayesian_probability "Bayesian probability") [prior](https://en.wikipedia.org/wiki/Prior_probability "Prior probability") [posterior](https://en.wikipedia.org/wiki/Posterior_probability "Posterior probability") [Credible interval](https://en.wikipedia.org/wiki/Credible_interval "Credible interval") [Bayes factor](https://en.wikipedia.org/wiki/Bayes_factor "Bayes factor") [Bayesian estimator](https://en.wikipedia.org/wiki/Bayes_estimator "Bayes estimator") [Maximum posterior estimator](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation "Maximum a posteriori estimation") |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | |
| | |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Pearson product-moment](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient "Pearson product-moment correlation coefficient") [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Confounding variable](https://en.wikipedia.org/wiki/Confounding "Confounding") [Coefficient of determination](https://en.wikipedia.org/wiki/Coefficient_of_determination "Coefficient of determination") |
| [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | [Errors and residuals](https://en.wikipedia.org/wiki/Errors_and_residuals "Errors and residuals") [Regression validation](https://en.wikipedia.org/wiki/Regression_validation "Regression validation") [Mixed effects models](https://en.wikipedia.org/wiki/Mixed_model "Mixed model") [Simultaneous equations models](https://en.wikipedia.org/wiki/Simultaneous_equations_model "Simultaneous equations model") [Multivariate adaptive regression splines (MARS)](https://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines "Multivariate adaptive regression splines")  [Template:Least squares and regression analysis](https://en.wikipedia.org/wiki/Template:Least_squares_and_regression_analysis "Template:Least squares and regression analysis") |
| [Linear regression](https://en.wikipedia.org/wiki/Linear_regression "Linear regression") | [Simple linear regression](https://en.wikipedia.org/wiki/Simple_linear_regression "Simple linear regression") [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares") [General linear model](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Bayesian regression](https://en.wikipedia.org/wiki/Bayesian_linear_regression "Bayesian linear regression") |
| Non-standard predictors | [Nonlinear regression](https://en.wikipedia.org/wiki/Nonlinear_regression "Nonlinear regression") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_regression "Nonparametric regression") [Semiparametric](https://en.wikipedia.org/wiki/Semiparametric_regression "Semiparametric regression") [Isotonic](https://en.wikipedia.org/wiki/Isotonic_regression "Isotonic regression") [Robust](https://en.wikipedia.org/wiki/Robust_regression "Robust regression") [Homoscedasticity and Heteroscedasticity](https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity "Homoscedasticity and heteroscedasticity") |
| [Generalized linear model](https://en.wikipedia.org/wiki/Generalized_linear_model "Generalized linear model") | [Exponential families](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Logistic (Bernoulli)](https://en.wikipedia.org/wiki/Logistic_regression "Logistic regression") / [Binomial](https://en.wikipedia.org/wiki/Binomial_regression "Binomial regression") / [Poisson regressions](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") |
| [Partition of variance](https://en.wikipedia.org/wiki/Partition_of_sums_of_squares "Partition of sums of squares") | [Analysis of variance (ANOVA, anova)](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [Analysis of covariance](https://en.wikipedia.org/wiki/Analysis_of_covariance "Analysis of covariance") [Multivariate ANOVA](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom_\(statistics\) "Degrees of freedom (statistics)") |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") / [multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") / [time-series](https://en.wikipedia.org/wiki/Time_series "Time series") / [survival analysis](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") | [Cohen's kappa](https://en.wikipedia.org/wiki/Cohen%27s_kappa "Cohen's kappa") [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Graphical model](https://en.wikipedia.org/wiki/Graphical_model "Graphical model") [Log-linear model](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") [McNemar's test](https://en.wikipedia.org/wiki/McNemar%27s_test "McNemar's test") [CochranâMantelâHaenszel statistics](https://en.wikipedia.org/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics "CochranâMantelâHaenszel statistics") |
| [Multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") | [Regression](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Manova](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Principal components](https://en.wikipedia.org/wiki/Principal_component_analysis "Principal component analysis") [Canonical correlation](https://en.wikipedia.org/wiki/Canonical_correlation "Canonical correlation") [Discriminant analysis](https://en.wikipedia.org/wiki/Linear_discriminant_analysis "Linear discriminant analysis") [Cluster analysis](https://en.wikipedia.org/wiki/Cluster_analysis "Cluster analysis") [Classification](https://en.wikipedia.org/wiki/Statistical_classification "Statistical classification") [Structural equation model](https://en.wikipedia.org/wiki/Structural_equation_modeling "Structural equation modeling") [Factor analysis](https://en.wikipedia.org/wiki/Factor_analysis "Factor analysis") [Multivariate distributions](https://en.wikipedia.org/wiki/Multivariate_distribution "Multivariate distribution") [Elliptical distributions](https://en.wikipedia.org/wiki/Elliptical_distribution "Elliptical distribution") [Normal](https://en.wikipedia.org/wiki/Multivariate_normal_distribution "Multivariate normal distribution") |
| [Time-series](https://en.wikipedia.org/wiki/Time_series "Time series") | |
| | |
| General | [Decomposition](https://en.wikipedia.org/wiki/Decomposition_of_time_series "Decomposition of time series") [Trend](https://en.wikipedia.org/wiki/Trend_estimation "Trend estimation") [Stationarity](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") [Seasonal adjustment](https://en.wikipedia.org/wiki/Seasonal_adjustment "Seasonal adjustment") [Exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing "Exponential smoothing") [Cointegration](https://en.wikipedia.org/wiki/Cointegration "Cointegration") [Structural break](https://en.wikipedia.org/wiki/Structural_break "Structural break") [Granger causality](https://en.wikipedia.org/wiki/Granger_causality "Granger causality") |
| Specific tests | [DickeyâFuller](https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test "DickeyâFuller test") [Johansen](https://en.wikipedia.org/wiki/Johansen_test "Johansen test") [Q-statistic (LjungâBox)](https://en.wikipedia.org/wiki/Ljung%E2%80%93Box_test "LjungâBox test") [DurbinâWatson](https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic "DurbinâWatson statistic") [BreuschâGodfrey](https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey_test "BreuschâGodfrey test") |
| [Time domain](https://en.wikipedia.org/wiki/Time_domain "Time domain") | [Autocorrelation (ACF)](https://en.wikipedia.org/wiki/Autocorrelation "Autocorrelation") [partial (PACF)](https://en.wikipedia.org/wiki/Partial_autocorrelation_function "Partial autocorrelation function") [Cross-correlation (XCF)](https://en.wikipedia.org/wiki/Cross-correlation "Cross-correlation") [ARMA model](https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model "Autoregressiveâmoving-average model") [ARIMA model (BoxâJenkins)](https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method "BoxâJenkins method") [Autoregressive conditional heteroskedasticity (ARCH)](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") [Vector autoregression (VAR)](https://en.wikipedia.org/wiki/Vector_autoregression "Vector autoregression") ([Autoregressive model (AR)](https://en.wikipedia.org/wiki/Autoregressive_model "Autoregressive model")) |
| [Frequency domain](https://en.wikipedia.org/wiki/Frequency_domain "Frequency domain") | [Spectral density estimation](https://en.wikipedia.org/wiki/Spectral_density_estimation "Spectral density estimation") [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") [Least-squares spectral analysis](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis "Least-squares spectral analysis") [Wavelet](https://en.wikipedia.org/wiki/Wavelet "Wavelet") [Whittle likelihood](https://en.wikipedia.org/wiki/Whittle_likelihood "Whittle likelihood") |
| [Survival](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
| [Survival function](https://en.wikipedia.org/wiki/Survival_function "Survival function") | [KaplanâMeier estimator (product limit)](https://en.wikipedia.org/wiki/Kaplan%E2%80%93Meier_estimator "KaplanâMeier estimator") [Proportional hazards models](https://en.wikipedia.org/wiki/Proportional_hazards_model "Proportional hazards model") [Accelerated failure time (AFT) model](https://en.wikipedia.org/wiki/Accelerated_failure_time_model "Accelerated failure time model") [First hitting time](https://en.wikipedia.org/wiki/First-hitting-time_model "First-hitting-time model") |
| [Hazard function](https://en.wikipedia.org/wiki/Failure_rate "Failure rate") | [NelsonâAalen estimator](https://en.wikipedia.org/wiki/Nelson%E2%80%93Aalen_estimator "NelsonâAalen estimator") |
| Test | [Log-rank test](https://en.wikipedia.org/wiki/Log-rank_test "Log-rank test") |
| [Applications](https://en.wikipedia.org/wiki/List_of_fields_of_application_of_statistics "List of fields of application of statistics") | |
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| [Social statistics](https://en.wikipedia.org/wiki/Social_statistics "Social statistics") | [Actuarial science](https://en.wikipedia.org/wiki/Actuarial_science "Actuarial science") [Census](https://en.wikipedia.org/wiki/Census "Census") [Crime statistics](https://en.wikipedia.org/wiki/Crime_statistics "Crime statistics") [Demography](https://en.wikipedia.org/wiki/Demographic_statistics "Demographic statistics") [Econometrics](https://en.wikipedia.org/wiki/Econometrics "Econometrics") [Jurimetrics](https://en.wikipedia.org/wiki/Jurimetrics "Jurimetrics") [National accounts](https://en.wikipedia.org/wiki/National_accounts "National accounts") [Official statistics](https://en.wikipedia.org/wiki/Official_statistics "Official statistics") [Population statistics](https://en.wikipedia.org/wiki/Population_statistics "Population statistics") [Psychometrics](https://en.wikipedia.org/wiki/Psychometrics "Psychometrics") |
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Central limit theorem
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[Add topic](https://en.wikipedia.org/wiki/Central_limit_theorem) |
| Readable Markdown | | | |
|---|---|
| [](https://en.wikipedia.org/wiki/File:IllustrationCentralTheorem.png) | |
| Type | [Theorem](https://en.wikipedia.org/wiki/Theorem "Theorem") |
| Field | [Probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") |
| Statement | The scaled sum of a sequence of [i.i.d. random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables") with finite positive [variance](https://en.wikipedia.org/wiki/Variance "Variance") converges in distribution to the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"). |
| Generalizations | [Lindeberg's CLT](https://en.wikipedia.org/wiki/Lindeberg%27s_condition "Lindeberg's condition") |
In [probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory"), the **central limit theorem** (**CLT**) states that, under appropriate conditions, the [distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") of a normalized version of the sample mean converges to a [standard normal distribution](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution "Normal distribution"). This holds even if the original variables themselves are not [normally distributed](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"). There are several versions of the CLT, each applying in the context of different conditions.
The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated in the 1920s.[\[1\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">[<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears. \(July_2023\)">page needed</span>]]</i>]</sup>-1)
In [statistics](https://en.wikipedia.org/wiki/Statistics "Statistics"), the CLT can be stated as: let  denote a [statistical sample](https://en.wikipedia.org/wiki/Sampling_\(statistics\) "Sampling (statistics)") of size  from a population with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") (average)  and finite positive [variance](https://en.wikipedia.org/wiki/Variance "Variance") , and let  denote the sample mean (which is itself a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable")). Then the [limit as  of the distribution](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_distribution "Convergence of random variables") of  is a normal distribution with mean  and variance .[\[2\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-2)
In other words, suppose that a large sample of [observations](https://en.wikipedia.org/wiki/Random_variate "Random variate") is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average ([arithmetic mean](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean")) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the [probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") of these averages will closely approximate a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be [independent and identically distributed](https://en.wikipedia.org/wiki/Independent_and_identically_distributed "Independent and identically distributed") (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution"), is the [de MoivreâLaplace theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem "De MoivreâLaplace theorem").
## Independent sequences
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=1 "Edit section: Independent sequences")\]
[](https://en.wikipedia.org/wiki/File:IllustrationCentralTheorem.png)
Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.[\[3\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-3)
Let  be a sequence of [i.i.d. random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables") having a distribution with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") given by  and finite [variance](https://en.wikipedia.org/wiki/Variance "Variance") given by  Suppose we are interested in the [sample average](https://en.wikipedia.org/wiki/Sample_mean "Sample mean")
By the [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers"), the sample average [converges almost surely](https://en.wikipedia.org/wiki/Almost_sure_convergence "Almost sure convergence") (and therefore also [converges in probability](https://en.wikipedia.org/wiki/Convergence_in_probability "Convergence in probability")) to the expected value  as 
The classical central limit theorem describes the size and the distributional form of the [stochastic](https://en.wiktionary.org/wiki/stochastic "wikt:stochastic") fluctuations around the deterministic number  during this convergence. More precisely, it states that as  gets larger, the distribution of the normalized mean , i.e. the difference between the sample average  and its limit  scaled by the factor , approaches the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with mean  and variance  For large enough  the distribution of  gets arbitrarily close to the normal distribution with mean  and variance 
The usefulness of the theorem is that the distribution of  approaches normality regardless of the shape of the distribution of the individual  Formally, the theorem can be stated as follows:
In the case  convergence in distribution means that the [cumulative distribution functions](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function") of  converge pointwise to the cdf of the  distribution: for every real number 
![{\\displaystyle \\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]=\\lim \_{n\\to \\infty }\\mathbb {P} \\left\[{\\frac {{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )}{\\sigma }}\\leq {\\frac {z}{\\sigma }}\\right\]=\\Phi \\left({\\frac {z}{\\sigma }}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/defd4cf70972fa6a76a8570fee6551f4cb7d70b8)
where  is the standard normal cdf evaluated at  The convergence is uniform in  in the sense that
![{\\displaystyle \\lim \_{n\\to \\infty }\\;\\sup \_{z\\in \\mathbb {R} }\\;\\left\|\\mathbb {P} \\left\[{\\sqrt {n}}({\\bar {X}}\_{n}-\\mu )\\leq z\\right\]-\\Phi \\left({\\frac {z}{\\sigma }}\\right)\\right\|=0~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835addcb3ec37594d1e9a6a78c0373a5e7b2eddc)
where  denotes the [supremum](https://en.wikipedia.org/wiki/Supremum "Supremum") (i.e. least upper bound) of the set.[\[5\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBauer2001199Theorem_30.13-5)
In this variant of the central limit theorem the random variables  have to be independent, but not necessarily identically distributed. The theorem also requires that random variables  have [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of some order , and that the rate of growth of these moments is limited by the Lyapunov condition given below.
**Lyapunov CLT[\[6\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995362-6)**âSuppose  is a sequence of independent random variables, each with finite expected value  and variance . Define
If for some , *Lyapunovâs condition*
![{\\displaystyle \\lim \_{n\\to \\infty }\\;{\\frac {1}{s\_{n}^{2+\\delta }}}\\,\\sum \_{i=1}^{n}\\operatorname {E} \\left\[\\left\|X\_{i}-\\mu \_{i}\\right\|^{2+\\delta }\\right\]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f06b7a3309fd45005cce7a4e0b15ca3758f662f5)
is satisfied, then a sum of  converges in distribution to a standard normal random variable, as  goes to infinity:
In practice it is usually easiest to check Lyapunov's condition for .
If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.
### Lindeberg (-Feller) CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=4 "Edit section: Lindeberg (-Feller) CLT")\]
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from [Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg") in 1920).
Suppose that for every ,
![{\\displaystyle \\lim \_{n\\to \\infty }{\\frac {1}{s\_{n}^{2}}}\\sum \_{i=1}^{n}\\operatorname {E} \\left\[(X\_{i}-\\mu \_{i})^{2}\\cdot \\mathbf {1} \_{\\left\\{\\left\|X\_{i}-\\mu \_{i}\\right\|\>\\varepsilon s\_{n}\\right\\}}\\right\]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd10d152dd578ac2a2fa674a084bd7b03b95b1b)
where  is the [indicator function](https://en.wikipedia.org/wiki/Indicator_function "Indicator function"). Then the distribution of the standardized sums
converges towards the standard normal distribution .
### CLT for the sum of a random number of random variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=5 "Edit section: CLT for the sum of a random number of random variables")\]
Rather than summing an integer number  of random variables and taking , the sum can be of a random number  of random variables, with conditions on . For example, the following theorem is Corollary 4 of Robbins (1948). It assumes that  is asymptotically normal (Robbins also developed other conditions that lead to the same result).
### Multidimensional CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=6 "Edit section: Multidimensional CLT")\]
Proofs that use characteristic functions can be extended to cases where each individual  is a [random vector](https://en.wikipedia.org/wiki/Random_vector "Random vector") in , with mean vector ![{\\textstyle {\\boldsymbol {\\mu }}=\\operatorname {E} \[\\mathbf {X} \_{i}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2931b016ce578d17578ee3cdffeb31852446873) and [covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix "Covariance matrix")  (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a [multivariate normal distribution](https://en.wikipedia.org/wiki/Multivariate_normal_distribution "Multivariate normal distribution").[\[9\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-vanderVaart-9) Summation of these vectors is done component-wise.
For  let
be independent random vectors. The sum of the random vectors  is
and their average is
Therefore,
![{\\displaystyle {\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}\\left\[\\mathbf {X} \_{i}-\\operatorname {E} \\left(\\mathbf {X} \_{i}\\right)\\right\]={\\frac {1}{\\sqrt {n}}}\\sum \_{i=1}^{n}(\\mathbf {X} \_{i}-{\\boldsymbol {\\mu }})={\\sqrt {n}}\\left({\\overline {\\mathbf {X} }}\_{n}-{\\boldsymbol {\\mu }}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d3c26b99d8a97339b688367c519d0e28f9541c)
The multivariate central limit theorem states that
 where the [covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix "Covariance matrix")  is equal to 
The multivariate central limit theorem can be proved using the [CramĂ©râWold theorem](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Wold_theorem "CramĂ©râWold theorem").[\[9\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-vanderVaart-9)
The rate of convergence is given by the following [BerryâEsseen](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem") type result:
It is unknown whether the factor  is necessary.[\[11\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-11)
## The generalized central limit theorem
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=7 "Edit section: The generalized central limit theorem")\]
The generalized central limit theorem (GCLT) was an effort of multiple mathematicians ([Sergei Bernstein](https://en.wikipedia.org/wiki/Sergei_Bernstein "Sergei Bernstein"), [Jarl Waldemar Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg"), [Paul LĂ©vy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul LĂ©vy (mathematician)"), [William Feller](https://en.wikipedia.org/wiki/William_Feller "William Feller"), [Andrey Kolmogorov](https://en.wikipedia.org/wiki/Andrey_Kolmogorov "Andrey Kolmogorov"), and others) over the period from 1920 to 1937.[\[12\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-12) The first published complete proof of the GCLT was in 1937 by Paul LĂ©vy in French.[\[13\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-13) An English language version of the complete proof of the GCLT is available in the translation of [Boris Vladimirovich Gnedenko](https://en.wikipedia.org/wiki/Boris_Vladimirovich_Gnedenko "Boris Vladimirovich Gnedenko") and Kolmogorov's 1954 book.[\[14\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-14)
The statement of the GCLT is as follows:[\[15\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-15)
**Statement of GCLT**âA non-degenerate random variable Z is [α\-stable](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution") for some 0 \< *α* †2 if and only if there is an independent, identically distributed sequence of random variables *X*1, *X*2, *X*3, ..., and constants *a**n* \> 0, *b**n* â â with  Here, 'â' means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy *F**n*(*y*) â *F*(*y*) at all continuity points of F.
In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a [stable distribution](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution").
## Dependent processes
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=8 "Edit section: Dependent processes")\]
### CLT under weak dependence
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=9 "Edit section: CLT under weak dependence")\]
A useful generalization of a sequence of independent, identically distributed random variables is a [mixing](https://en.wikipedia.org/wiki/Mixing_\(mathematics\) "Mixing (mathematics)") random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially [strong mixing](https://en.wikipedia.org/wiki/Mixing_\(mathematics\)#Mixing_in_stochastic_processes "Mixing (mathematics)") (also called α-mixing) defined by  where  is so-called [strong mixing coefficient](https://en.wikipedia.org/wiki/Mixing_\(mathematics\)#Mixing_in_stochastic_processes "Mixing (mathematics)").
A simplified formulation of the central limit theorem under strong mixing is:[\[16\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995Theorem_27.4-16)
In fact,
where the series converges absolutely.
The assumption  cannot be omitted, since the asymptotic normality fails for  where  are another [stationary sequence](https://en.wikipedia.org/wiki/Stationary_sequence "Stationary sequence").
There is a stronger version of the theorem:[\[17\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Sect._7.7\(c\),_Theorem_7.8-17) the assumption ![{\\textstyle \\operatorname {E} \\left\[X\_{n}^{12}\\right\]\<\\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/657baea1ae3ea930c0d5057a69e2353a155a0df4) is replaced with ![{\\textstyle \\operatorname {E} \\left\[{\\left\|X\_{n}\\right\|}^{2+\\delta }\\right\]\<\\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/940680c6ace3f0b2a962e5e31fcc79b6e4f28f13), and the assumption  is replaced with
Existence of such  ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see ([Bradley 2007](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBradley2007)).
### Martingale difference CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=10 "Edit section: Martingale difference CLT")\]
**Theorem**âLet a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)")  satisfy
then  converges in distribution to  as .[\[18\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4-18)[\[19\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEBillingsley1995Theorem_35.12-19)
### Proof of classical CLT
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=12 "Edit section: Proof of classical CLT")\]
The central limit theorem has a proof using [characteristic functions](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)").[\[20\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-20) It is similar to the proof of the (weak) [law of large numbers](https://en.wikipedia.org/wiki/Proof_of_the_law_of_large_numbers "Proof of the law of large numbers").
Assume  are independent and identically distributed random variables, each with mean  and finite variance . The sum  has [mean](https://en.wikipedia.org/wiki/Linearity_of_expectation "Linearity of expectation")  and [variance](https://en.wikipedia.org/wiki/Variance#Sum_of_uncorrelated_variables_\(Bienaym%C3%A9_formula\) "Variance") . Consider the random variable
where in the last step we defined the new random variables , each with zero mean and unit variance (). The [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)") of  is given by
![{\\displaystyle {\\begin{aligned}\\varphi \_{Z\_{n}}\\!(t)=\\varphi \_{\\sum \_{i=1}^{n}{{\\frac {1}{\\sqrt {n}}}Y\_{i}}}\\!(t)\\ &=\\ \\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\varphi \_{Y\_{2}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\cdots \\varphi \_{Y\_{n}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\\\\[1ex\]&=\\ \\left\[\\varphi \_{Y\_{1}}\\!\\!\\left({\\frac {t}{\\sqrt {n}}}\\right)\\right\]^{n},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b0081e06952eb33ecd2d847e4c8bf6c41449bd)
where in the last step we used the fact that all of the  are identically distributed. The characteristic function of  is, by [Taylor's theorem](https://en.wikipedia.org/wiki/Taylor%27s_theorem "Taylor's theorem"), 
where  is "[little o notation](https://en.wikipedia.org/wiki/Little-o_notation "Little-o notation")" for some function of  that goes to zero more rapidly than . By the limit of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") (), the characteristic function of  equals
All of the higher order terms vanish in the limit . The right hand side equals the characteristic function of a standard normal distribution , which implies through [LĂ©vy's continuity theorem](https://en.wikipedia.org/wiki/L%C3%A9vy_continuity_theorem "LĂ©vy continuity theorem") that the distribution of  will approach  as . Therefore, the [sample average](https://en.wikipedia.org/wiki/Sample_mean "Sample mean")
is such that
converges to the normal distribution , from which the central limit theorem follows.
### Convergence to the limit
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=13 "Edit section: Convergence to the limit")\]
The central limit theorem gives only an [asymptotic distribution](https://en.wikipedia.org/wiki/Asymptotic_distribution "Asymptotic distribution"). As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The convergence in the central limit theorem is [uniform](https://en.wikipedia.org/wiki/Uniform_convergence "Uniform convergence") because the limiting cumulative distribution function is continuous. If the third central [moment](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") ![{\\textstyle \\operatorname {E} \\left\[(X\_{1}-\\mu )^{3}\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389b355cd56db15cdf7e88c8b0aff830a381726f) exists and is finite, then the speed of convergence is at least on the order of  (see [BerryâEsseen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem")). [Stein's method](https://en.wikipedia.org/wiki/Stein%27s_method "Stein's method")[\[21\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-stein1972-21) can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.[\[22\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-22)
The convergence to the normal distribution is monotonic, in the sense that the [entropy](https://en.wikipedia.org/wiki/Information_entropy "Information entropy") of  increases [monotonically](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") to that of the normal distribution.[\[23\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-ABBN-23)
The central limit theorem applies in particular to sums of independent and identically distributed [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable"). A sum of [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable") is still a [discrete random variable](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable"), so that we are confronted with a sequence of [discrete random variables](https://en.wikipedia.org/wiki/Discrete_random_variable "Discrete random variable") whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution")). This means that if we build a [histogram](https://en.wikipedia.org/wiki/Histogram "Histogram") of the realizations of the sum of n independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity; this relation is known as [de MoivreâLaplace theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem "De MoivreâLaplace theorem"). The [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution") article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
### Common misconceptions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=14 "Edit section: Common misconceptions")\]
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.[\[24\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-26) These include:
- The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of [iid](https://en.wikipedia.org/wiki/Iid "Iid") random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a [sampling distribution](https://en.wikipedia.org/wiki/Sampling_distribution "Sampling distribution") formed from different values of means (or sums) of such random variables.
- The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the [GlivenkoâCantelli theorem](https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem "GlivenkoâCantelli theorem").
- The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,[\[27\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-27) allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See [Z-test](https://en.wikipedia.org/wiki/Z-test "Z-test") for where the approximation holds.
### Relation to the law of large numbers
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=15 "Edit section: Relation to the law of large numbers")\]
The [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers") as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of Sn as n approaches infinity?" In mathematical analysis, [asymptotic series](https://en.wikipedia.org/wiki/Asymptotic_series "Asymptotic series") are one of the most popular tools employed to approach such questions.
Suppose we have an asymptotic expansion of :
Dividing both parts by *Ï*1(*n*) and taking the limit will produce *a*1, the coefficient of the highest-order term in the expansion, which represents the rate at which *f*(*n*) changes in its leading term.
Informally, one can say: "*f*(*n*) grows approximately as *a*1*Ï*1(*n*)". Taking the difference between *f*(*n*) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about *f*(*n*):
Here one can say that the difference between the function and its approximation grows approximately as *a*2*Ï*2(*n*). The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.
Informally, something along these lines happens when the sum, Sn, of independent identically distributed random variables, *X*1, ..., *Xn*, is studied in classical probability theory.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] If each Xi has finite mean ÎŒ, then by the law of large numbers, â *Sn*/*n*â â *ÎŒ*.[\[28\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-28) If in addition each Xi has finite variance *Ï*2, then by the central limit theorem,
where Ο is distributed as *N*(0,*Ï*2). This provides values of the first two constants in the informal expansion
In the case where the Xi do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
or informally
Distributions Î which can arise in this way are called *[stable](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution")*.[\[29\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-29) Clearly, the normal distribution is stable, but there are also other stable distributions, such as the [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution "Cauchy distribution"), for which the mean or variance are not defined. The scaling factor bn may be proportional to nc, for any *c* â„ â 1/2â ; it may also be multiplied by a [slowly varying function](https://en.wikipedia.org/wiki/Slowly_varying_function "Slowly varying function") of n.[\[30\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Uchaikin-30)[\[31\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-31)
The [law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm "Law of the iterated logarithm") specifies what is happening "in between" the [law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers") and the central limit theorem. Specifically it says that the normalizing function â*n* log log *n*, intermediate in size between n of the law of large numbers and â*n* of the central limit theorem, provides a non-trivial limiting behavior.
### Alternative statements of the theorem
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=16 "Edit section: Alternative statements of the theorem")\]
The [density](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") of the sum of two or more independent variables is the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov[\[32\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-32) for a particular local limit theorem for sums of [independent and identically distributed random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables").
#### Characteristic functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=18 "Edit section: Characteristic functions")\]
Since the [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_\(probability_theory\) "Characteristic function (probability theory)") of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
An equivalent statement can be made about [Fourier transforms](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform"), since the characteristic function is essentially a Fourier transform.
### Calculating the variance
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=19 "Edit section: Calculating the variance")\]
Let Sn be the sum of n random variables. Many central limit theorems provide conditions such that Sn/âVar(Sn) converges in distribution to *N*(0,1) (the normal distribution with mean 0, variance 1) as n â â. In some cases, it is possible to find a constant *Ï*2 and function f(n) such that Sn/(Ïânâ
f(n)) converges in distribution to *N*(0,1) as nâ â.
### Products of positive random variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=21 "Edit section: Products of positive random variables")\]
The [logarithm](https://en.wikipedia.org/wiki/Logarithm "Logarithm") of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a [log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution "Log-normal distribution"). Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different [random](https://en.wikipedia.org/wiki/Random "Random") factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called [Gibrat's law](https://en.wikipedia.org/wiki/Gibrat%27s_law "Gibrat's law").
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.[\[34\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Rempala-34)
## Beyond the classical framework
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=22 "Edit section: Beyond the classical framework")\]
Asymptotic normality, that is, [convergence](https://en.wikipedia.org/wiki/Convergence_in_distribution "Convergence in distribution") to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
**Theorem**âThere exists a sequence *Δn* â 0 for which the following holds. Let *n* â„ 1, and let random variables *X*1, ..., *Xn* have a [log-concave](https://en.wikipedia.org/wiki/Logarithmically_concave_function "Logarithmically concave function") [joint density](https://en.wikipedia.org/wiki/Joint_density_function "Joint density function") f such that *f*(*x*1, ..., *xn*) = *f*(\|*x*1\|, ..., \|*xn*\|) for all *x*1, ..., *xn*, and E(*X*2
*k*) = 1 for all *k* = 1, ..., *n*. Then the distribution of
is Δn\-close to  in the [total variation distance](https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures "Total variation distance of probability measures").[\[35\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2007Theorem_1.2-35)
These two Δn\-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Another example: *f*(*x*1, ..., *xn*) = const · exp(â(\|*x*1\|*α* + ⯠+ \|*xn*\|*α*)*ÎČ*) where *α* \> 1 and *αÎČ* \> 1. If *ÎČ* = 1 then *f*(*x*1, ..., *xn*) factorizes into const · exp (â\|*x*1\|*α*) ⊠exp(â\|*xn*\|*α*), which means *X*1, ..., *Xn* are independent. In general, however, they are dependent.
The condition *f*(*x*1, ..., *xn*) = *f*(\|*x*1\|, ..., \|*xn*\|) ensures that *X*1, ..., *Xn* are of zero mean and [uncorrelated](https://en.wikipedia.org/wiki/Uncorrelated "Uncorrelated");\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] still, they need not be independent, nor even [pairwise independent](https://en.wikipedia.org/wiki/Pairwise_independence "Pairwise independence").\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] By the way, pairwise independence cannot replace independence in the classical central limit theorem.[\[36\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEDurrett2004Section_2.4,_Example_4.5-36)
Here is a [BerryâEsseen](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem") type result.
**Theorem**âLet *X*1, ..., *Xn* satisfy the assumptions of the previous theorem, then[\[37\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2008Theorem_1-37)
for all *a* \< *b*; here C is a [universal (absolute) constant](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant"). Moreover, for every *c*1, ..., *cn* â **R** such that *c*2
1 + ⯠+ *c*2
*n* = 1,
The distribution of â *X*1 + ⯠+ *Xn*/â*n*â need not be approximately normal (in fact, it can be uniform).[\[38\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEKlartag2007Theorem_1.1-38) However, the distribution of *c*1*X*1 + ⯠+ *cnXn* is close to  (in the total variation distance) for most vectors (*c*1, ..., *cn*) according to the uniform distribution on the sphere *c*2
1 + ⯠+ *c*2
*n* = 1.
### Lacunary trigonometric series
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=24 "Edit section: Lacunary trigonometric series")\]
**Theorem ([Salem](https://en.wikipedia.org/wiki/Rapha%C3%ABl_Salem "RaphaĂ«l Salem")â[Zygmund](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund"))**âLet U be a random variable distributed uniformly on (0,2Ï), and *Xk* = *rk* cos(*nkU* + *ak*), where
- nk satisfy the lacunarity condition: there exists *q* \> 1 such that *n**k* + 1 â„ *qn**k* for all k,
- rk are such that
- 0 †*a**k* \< 2Ï.
Then[\[39\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Zygmund-39)[\[40\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEGaposhkin1966Theorem_2.1.13-40)
converges in distribution to .
**Theorem**âLet *A*1, ..., *A**n* be independent random points on the plane **R**2 each having the two-dimensional standard normal distribution. Let Kn be the [convex hull](https://en.wikipedia.org/wiki/Convex_hull "Convex hull") of these points, and Xn the area of Kn Then[\[41\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.1-41)
 converges in distribution to  as n tends to infinity.
The same also holds in all dimensions greater than 2.
The [polytope](https://en.wikipedia.org/wiki/Convex_polytope "Convex polytope") Kn is called a Gaussian [random polytope](https://en.wikipedia.org/wiki/Random_polytope "Random polytope").
A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.[\[42\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.2-42)
### Linear functions of orthogonal matrices
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=26 "Edit section: Linear functions of orthogonal matrices")\]
A linear function of a matrix **M** is a linear combination of its elements (with given coefficients), **M** ⊠tr(**AM**) where **A** is the matrix of the coefficients; see [Trace (linear algebra)\#Inner product](https://en.wikipedia.org/wiki/Trace_\(linear_algebra\)#Inner_product "Trace (linear algebra)").
A random [orthogonal matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix "Orthogonal matrix") is said to be distributed uniformly, if its distribution is the normalized [Haar measure](https://en.wikipedia.org/wiki/Haar_measure "Haar measure") on the [orthogonal group](https://en.wikipedia.org/wiki/Orthogonal_group "Orthogonal group") O(*n*,**R**); see [Rotation matrix\#Uniform random rotation matrices](https://en.wikipedia.org/wiki/Rotation_matrix#Uniform_random_rotation_matrices "Rotation matrix").
**Theorem**âLet **M** be a random orthogonal *n* Ă *n* matrix distributed uniformly, and **A** a fixed *n* Ă *n* matrix such that tr(**AA**\*) = *n*, and let *X* = tr(**AM**). Then[\[43\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Meckes-43) the distribution of X is close to  in the total variation metric up to\[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")*\] â 2â3/*n* â 1â .
**Theorem**âLet random variables *X*1, *X*2, ... â *L*2(Ω) be such that *Xn* â 0 [weakly](https://en.wikipedia.org/wiki/Weak_convergence_\(Hilbert_space\) "Weak convergence (Hilbert space)") in *L*2(Ω) and *X*
*n* â 1 weakly in *L*1(Ω). Then there exist integers *n*1 \< *n*2 \< ⯠such that
converges in distribution to  as k tends to infinity.[\[44\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEGaposhkin1966Sect._1.5-44)
### Random walk on a crystal lattice
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=28 "Edit section: Random walk on a crystal lattice")\]
The central limit theorem may be established for the simple [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.[\[45\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-45)[\[46\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-46)
## Applications and examples
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=29 "Edit section: Applications and examples")\]
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample [statistics](https://en.wikipedia.org/wiki/Statistic "Statistic") to the normal distribution in controlled experiments.
[](https://en.wikipedia.org/wiki/File:Dice_sum_central_limit_theorem.svg)
Comparison of probability density functions *p*(*k*) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
[](https://en.wikipedia.org/wiki/File:Empirical_CLT_-_Figure_-_040711.jpg)
This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the [chi-squared](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test "Pearson's chi-squared test") values that quantify the goodness of the fit (the fit is good if the reduced [chi-squared](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test "Pearson's chi-squared test") value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/â*n*), which is called the standard deviation of the mean (since it refers to the spread of sample means).
[](https://en.wikipedia.org/wiki/File:Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg)
Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.
[Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis"), and in particular [ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares"), specifies that a [dependent variable](https://en.wikipedia.org/wiki/Dependent_variable "Dependent variable") depends according to some function upon one or more [independent variables](https://en.wikipedia.org/wiki/Independent_variable "Independent variable"), with an additive [error term](https://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics "Errors and residuals in statistics"). Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.
### Other illustrations
\[[edit](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&action=edit§ion=31 "Edit section: Other illustrations")\]
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.[\[47\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Marasinghe-47)
Dutch mathematician [Henk Tijms](https://en.wikipedia.org/wiki/Henk_Tijms "Henk Tijms") writes:[\[48\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Tijms-48)
> The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician [Abraham de Moivre](https://en.wikipedia.org/wiki/Abraham_de_Moivre "Abraham de Moivre") who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician [Pierre-Simon Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace") rescued it from obscurity in his monumental work *Théorie analytique des probabilités*, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.
Sir [Francis Galton](https://en.wikipedia.org/wiki/Francis_Galton "Francis Galton") described the Central Limit Theorem in this way:[\[49\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-49)
> I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by [George PĂłlya](https://en.wikipedia.org/wiki/George_P%C3%B3lya "George PĂłlya") in 1920 in the title of a paper.[\[50\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Polya1920-50)[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) PĂłlya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word *central* in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) The abstract of the paper *On the central limit theorem of calculus of probability and the problem of moments* by PĂłlya[\[50\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Polya1920-50) in 1920 translates as follows.
> The occurrence of the Gaussian probability density 1 = *e*â*x*2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by [Liapounoff](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov"). ...
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as [Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy "Augustin-Louis Cauchy")'s, [Bessel](https://en.wikipedia.org/wiki/Friedrich_Bessel "Friedrich Bessel")'s and [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "Siméon Denis Poisson")'s contributions, is provided by Hald.[\[52\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Hald-52) Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by [von Mises](https://en.wikipedia.org/wiki/Richard_von_Mises "Richard von Mises"), [Pólya](https://en.wikipedia.org/wiki/George_P%C3%B3lya "George Pólya"), [Lindeberg](https://en.wikipedia.org/wiki/Jarl_Waldemar_Lindeberg "Jarl Waldemar Lindeberg"), [Lévy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul Lévy (mathematician)"), and [Cramér](https://en.wikipedia.org/wiki/Harald_Cram%C3%A9r "Harald Cramér") during the 1920s, are given by Hans Fischer.[\[53\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2-53) Le Cam describes a period around 1935.[\[51\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-LC1986-51) Bernstein[\[54\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-Bernstein-54) presents a historical discussion focusing on the work of [Pafnuty Chebyshev](https://en.wikipedia.org/wiki/Pafnuty_Chebyshev "Pafnuty Chebyshev") and his students [Andrey Markov](https://en.wikipedia.org/wiki/Andrey_Markov "Andrey Markov") and [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") that led to the first proofs of the CLT in a general setting.
A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of [Alan Turing](https://en.wikipedia.org/wiki/Alan_Turing "Alan Turing")'s 1934 Fellowship Dissertation for [King's College](https://en.wikipedia.org/wiki/King%27s_College,_Cambridge "King's College, Cambridge") at the [University of Cambridge](https://en.wikipedia.org/wiki/University_of_Cambridge "University of Cambridge"). Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.[\[55\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-55)
- [Asymptotic equipartition property](https://en.wikipedia.org/wiki/Asymptotic_equipartition_property "Asymptotic equipartition property")
- [Asymptotic distribution](https://en.wikipedia.org/wiki/Asymptotic_distribution "Asymptotic distribution")
- [Bates distribution](https://en.wikipedia.org/wiki/Bates_distribution "Bates distribution")
- [Benford's law](https://en.wikipedia.org/wiki/Benford%27s_law "Benford's law") â result of extension of CLT to product of random variables.
- [BerryâEsseen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem "BerryâEsseen theorem")
- [Central limit theorem for directional statistics](https://en.wikipedia.org/wiki/Central_limit_theorem_for_directional_statistics "Central limit theorem for directional statistics") â Central limit theorem applied to the case of directional statistics
- [Delta method](https://en.wikipedia.org/wiki/Delta_method "Delta method") â to compute the limit distribution of a function of a random variable.
- [ErdĆsâKac theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem "ErdĆsâKac theorem") â connects the number of prime factors of an integer with the normal probability distribution
- [FisherâTippettâGnedenko theorem](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem "FisherâTippettâGnedenko theorem") â limit theorem for extremum values (such as max{*Xn*})
- [IrwinâHall distribution](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution "IrwinâHall distribution")
- [Markov chain central limit theorem](https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem "Markov chain central limit theorem")
- [Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution")
- [Tweedie convergence theorem](https://en.wikipedia.org/wiki/Tweedie_distribution "Tweedie distribution") â a theorem that can be considered to bridge between the central limit theorem and the [Poisson convergence theorem](https://en.wikipedia.org/wiki/Poisson_convergence_theorem "Poisson convergence theorem")[\[56\]](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_note-J%C3%B8rgensen-1997-56)
- [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem")
1. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">[<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears. \(July_2023\)">page needed</span>]]</i>]</sup>_1-0)** [Fischer (2011)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFFischer2011), p. \[*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")*\].
2. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-2)**
Montgomery, Douglas C.; Runger, George C. (2014). *Applied Statistics and Probability for Engineers* (6th ed.). Wiley. p. 241. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781118539712](https://en.wikipedia.org/wiki/Special:BookSources/9781118539712 "Special:BookSources/9781118539712")
.
3. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-3)**
Rouaud, Mathieu (2013). [*Probability, Statistics and Estimation*](http://www.incertitudes.fr/book.pdf) (PDF). p. 10. [Archived](https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf) (PDF) from the original on 2022-10-09.
4. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995357_4-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), p. 357.
5. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBauer2001199Theorem_30.13_5-0)** [Bauer (2001)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBauer2001), p. 199, Theorem 30.13.
6. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995362_6-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), p. 362.
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Robbins, Herbert (1948). ["The asymptotic distribution of the sum of a random number of random variables"](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-12/The-asymptotic-distribution-of-the-sum-of-a-random-number/bams/1183513324.full). *Bull. Amer. Math. Soc*. **54** (12): 1151â1161\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0002-9904-1948-09142-X](https://doi.org/10.1090%2FS0002-9904-1948-09142-X).
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Chen, Louis H.Y.; Goldstein, Larry; Shao, Qi-Man (2011). *Normal Approximation by Stein's Method*. Berlin Heidelberg: Springer-Verlag. pp. 270â271\.
9. ^ [***a***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-vanderVaart_9-0) [***b***](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-vanderVaart_9-1)
van der Vaart, A.W. (1998). *Asymptotic statistics*. New York, NY: Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-49603-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-49603-2 "Special:BookSources/978-0-521-49603-2")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [98015176](https://lccn.loc.gov/98015176).
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[OâDonnell, Ryan](https://en.wikipedia.org/wiki/Ryan_O%27Donnell_\(computer_scientist\) "Ryan O'Donnell (computer scientist)") (2014). ["Theorem 5.38"](https://web.archive.org/web/20190408054104/http://www.contrib.andrew.cmu.edu/~ryanod/?p=866). Archived from [the original](http://www.contrib.andrew.cmu.edu/~ryanod/?p=866) on 2019-04-08. Retrieved 2017-10-18.
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Bentkus, V. (2005). "A Lyapunov-type bound in ". *Theory Probab. Appl*. **49** (2): 311â323\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1137/S0040585X97981123](https://doi.org/10.1137%2FS0040585X97981123).
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Le Cam, L. (February 1986). "The Central Limit Theorem around 1935". *Statistical Science*. **1** (1): 78â91\. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2245503](https://www.jstor.org/stable/2245503).
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LĂ©vy, Paul (1937). *Theorie de l'addition des variables aleatoires* \[*Combination theory of unpredictable variables*\] (in French). Paris: Gauthier-Villars.
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Gnedenko, Boris Vladimirovich; Kologorov, AndreÄ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968). *Limit distributions for sums of independent random variables*. Reading, MA: Addison-wesley.
15. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-15)**
Nolan, John P. (2020). [*Univariate stable distributions, Models for Heavy Tailed Data*](https://doi.org/10.1007/978-3-030-52915-4). Springer Series in Operations Research and Financial Engineering. Switzerland: Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-52915-4](https://doi.org/10.1007%2F978-3-030-52915-4). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-030-52914-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-52914-7 "Special:BookSources/978-3-030-52914-7")
. [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [226648987](https://api.semanticscholar.org/CorpusID:226648987).
16. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995Theorem_27.4_16-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), Theorem 27.4.
17. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Sect._7.7\(c\),_Theorem_7.8_17-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Sect. 7.7(c), Theorem 7.8.
18. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4_18-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Sect. 7.7, Theorem 7.4.
19. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEBillingsley1995Theorem_35.12_19-0)** [Billingsley (1995)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFBillingsley1995), Theorem 35.12.
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Lemons, Don (2003). [*An Introduction to Stochastic Processes in Physics*](https://jhupbooks.press.jhu.edu/content/introduction-stochastic-processes-physics). Johns Hopkins University Press. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.56021/9780801868665](https://doi.org/10.56021%2F9780801868665). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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[Stein, C.](https://en.wikipedia.org/wiki/Charles_Stein_\(statistician\) "Charles Stein (statistician)") (1972). ["A bound for the error in the normal approximation to the distribution of a sum of dependent random variables"](https://projecteuclid.org/euclid.bsmsp/1200514239). *Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability*. **6** (2): 583â602\. [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0402873](https://mathscinet.ams.org/mathscinet-getitem?mr=0402873). [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0278\.60026](https://zbmath.org/?format=complete&q=an:0278.60026).
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Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). *Normal approximation by Stein's method*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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23. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-ABBN_23-0)**
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25. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-25)** Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, *American Educational Research Association* lecture 19 April 1995
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27. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-27)**
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28. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-28)**
Rosenthal, Jeffrey Seth (2000). *A First Look at Rigorous Probability Theory*. World Scientific. Theorem 5.3.4, p. 47. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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29. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-29)**
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30. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Uchaikin_30-0)**
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33. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-33)**
Hew, Patrick Chisan (2017). "Asymptotic distribution of rewards accumulated by alternating renewal processes". *Statistics and Probability Letters*. **129**: 355â359\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.spl.2017.06.027](https://doi.org/10.1016%2Fj.spl.2017.06.027).
34. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Rempala_34-0)**
Rempala, G.; Wesolowski, J. (2002). ["Asymptotics of products of sums and *U*\-statistics"](https://projecteuclid.org/journals/electronic-communications-in-probability/volume-7/issue-none/Asymptotics-for-Products-of-Sums-and-U-statistics/10.1214/ECP.v7-1046.pdf) (PDF). *Electronic Communications in Probability*. **7**: 47â54\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1214/ecp.v7-1046](https://doi.org/10.1214%2Fecp.v7-1046).
35. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2007Theorem_1.2_35-0)** [Klartag (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2007), Theorem 1.2.
36. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEDurrett2004Section_2.4,_Example_4.5_36-0)** [Durrett (2004)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFDurrett2004), Section 2.4, Example 4.5.
37. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2008Theorem_1_37-0)** [Klartag (2008)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2008), Theorem 1.
38. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEKlartag2007Theorem_1.1_38-0)** [Klartag (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFKlartag2007), Theorem 1.1.
39. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Zygmund_39-0)**
[Zygmund, Antoni](https://en.wikipedia.org/wiki/Antoni_Zygmund "Antoni Zygmund") (2003) \[1959\]. [*Trigonometric Series*](https://en.wikipedia.org/wiki/Trigonometric_Series "Trigonometric Series"). Cambridge University Press. vol. II, sect. XVI.5, Theorem 5-5. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-521-89053-5](https://en.wikipedia.org/wiki/Special:BookSources/0-521-89053-5 "Special:BookSources/0-521-89053-5")
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40. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEGaposhkin1966Theorem_2.1.13_40-0)** [Gaposhkin (1966)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFGaposhkin1966), Theorem 2.1.13.
41. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.1_41-0)** [BĂĄrĂĄny & Vu (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFB%C3%A1r%C3%A1nyVu2007), Theorem 1.1.
42. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEB%C3%A1r%C3%A1nyVu2007Theorem_1.2_42-0)** [BĂĄrĂĄny & Vu (2007)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFB%C3%A1r%C3%A1nyVu2007), Theorem 1.2.
43. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Meckes_43-0)**
[Meckes, Elizabeth](https://en.wikipedia.org/wiki/Elizabeth_Meckes "Elizabeth Meckes") (2008). "Linear functions on the classical matrix groups". *Transactions of the American Mathematical Society*. **360** (10): 5355â5366\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0509441](https://arxiv.org/abs/math/0509441). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0002-9947-08-04444-9](https://doi.org/10.1090%2FS0002-9947-08-04444-9). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [11981408](https://api.semanticscholar.org/CorpusID:11981408).
44. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEGaposhkin1966Sect._1.5_44-0)** [Gaposhkin (1966)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFGaposhkin1966), Sect. 1.5.
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52. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Hald_52-0)**
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53. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2_53-0)** [Fischer (2011)](https://en.wikipedia.org/wiki/Central_limit_theorem#CITEREFFischer2011), Chapter 2; Chapter 5.2.
54. **[^](https://en.wikipedia.org/wiki/Central_limit_theorem#cite_ref-Bernstein_54-0)**
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- Fischer, Hans (2011). [*A History of the Central Limit Theorem: From Classical to Modern Probability Theory*](http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf) (PDF). Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-0-387-87857-7](https://doi.org/10.1007%2F978-0-387-87857-7). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-87856-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-87856-0 "Special:BookSources/978-0-387-87856-0")
. [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [2743162](https://mathscinet.ams.org/mathscinet-getitem?mr=2743162). [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [1226\.60004](https://zbmath.org/?format=complete&q=an:1226.60004). [Archived](https://web.archive.org/web/20171031171033/http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf) (PDF) from the original on 2017-10-31.
- Gaposhkin, V. F. (1966). "Lacunary series and independent functions". *Russian Mathematical Surveys*. **21** (6): 1â82\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1966RuMaS..21....1G](https://ui.adsabs.harvard.edu/abs/1966RuMaS..21....1G). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1070/RM1966v021n06ABEH001196](https://doi.org/10.1070%2FRM1966v021n06ABEH001196). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [250833638](https://api.semanticscholar.org/CorpusID:250833638).
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- Klartag, Bo'az (2007). "A central limit theorem for convex sets". *Inventiones Mathematicae*. **168** (1): 91â131\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0605014](https://arxiv.org/abs/math/0605014). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2007InMat.168...91K](https://ui.adsabs.harvard.edu/abs/2007InMat.168...91K). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s00222-006-0028-8](https://doi.org/10.1007%2Fs00222-006-0028-8). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119169773](https://api.semanticscholar.org/CorpusID:119169773).
- Klartag, Bo'az (2008). "A BerryâEsseen type inequality for convex bodies with an unconditional basis". *Probability Theory and Related Fields*. **145** (1â2\): 1â33\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[0705\.0832](https://arxiv.org/abs/0705.0832). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s00440-008-0158-6](https://doi.org/10.1007%2Fs00440-008-0158-6). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [10163322](https://api.semanticscholar.org/CorpusID:10163322).
- [Central Limit Theorem](https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/central-limit-theorem) at Khan Academy
- ["Central limit theorem"](https://www.encyclopediaofmath.org/index.php?title=Central_limit_theorem). *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*. [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"). 2001 \[1994\].
- [Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Central Limit Theorem"](https://mathworld.wolfram.com/CentralLimitTheorem.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
- [A music video demonstrating the central limit theorem with a Galton board](https://www.mctague.org/carl/blog/2021/04/23/central-limit-theorem/) by Carl McTague |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Central_limit_theorem s443 |