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Calculated Shard: 31 (from laksa038)

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curl -X POST \
  'http://laksa031.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.coursesidekick.com/statistics/69381\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.coursesidekick.com/statistics/69381\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.coursesidekick.com\/statistics\/69381","crawl_time":1774149769,"first_indexed_time":1682554343,"http_code":200,"src_unparsed":"com,coursesidekick!www,\/statistics\/69381 s443","src_root_hash":"2196228317939566631","history_drop_reason":null,"meta_title":"Markov Chains: Periodicity, Recurrence, and Transience Explained - Course Sidekick","meta_descriptions":["Statistics document from The University of Sydney, 19 pages, Week 3 Markov chains: Periodicity, recurrence and transience, positive and null recurrences 7 Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have (n) p00 = P (Xn = 0 | X0 = 0) 6= 0 (="],"attrs_boilerpipe_text":"Week 3\nMarkov chains:\nPeriodicity, recurrence and transience, positive and\nnull recurrences\n7\nMarkov chain: periodicity\nConsider the Markov chain shown below:\nThere is a periodic pattern in this chain. Indeed, we have\np\n(\nn\n)\n00\n=\nP\n(\nX\nn\n= 0\n|\nX\n0\n= 0)\n6\n= 0 (= 1)\n,\nif\nn\n= 3\n,\n6\n,\n9\n,\n· · ·\n,\np\n(\nn\n)\n00\n=\nP\n(\nX\nn\n= 0\n|\nX\n0\n= 0) = 0\n,\nif\nn\n6\n= 3\n,\n6\n,\n9\n,\n· · ·\n.\nSuch a state is called a\nperiodic\nstate with period\nd\n0\n= 3.\nThe\nperiod\nof a state\ni\n, denoted by\nd\ni\n, is defined as the greatest common\ndivisor (gcd) of all\nn\n≥\n1 for which\np\n(\nn\n)\nii\n>\n0, i.e.,\nd\ni\n= gcd\n{\nn\n:\np\n(\nn\n)\nii\n>\n0\n}\n.\nIf\np\n(\nn\n)\nii\n= 0, for all\nn\n≥\n1, then we let\nd\ni\n= 0 (or\nd\ni\n=\n∞\n).\n•\nIf\nd\ni\n>\n1, we say that state\ni\nis\nperiodic\n.\nIn this case,\np\n(\nkd\ni\n)\nii\n6\n= 0 for all\nk\n≥\nk\n0\n, where\nk\n0\n>\n1 is an integer, and\np\n(\nn\n)\nii\n= 0 when\nn\n6\n=\nkd\ni\n. Note that it happens\np\n(\nd\ni\n)\nii\n= 0.\n•\nIf\nd\ni\n= 1, we say that state\ni\nis\naperiodic\n.\nIn mathematics, the greatest common divisor (gcd) of two or more in-\ntegers, which are not all zero, is the largest positive integer that divides\neach of the integers. For example, the gcd of 8 and 12 is 4.\n31\nWant to read all 19 pages?\nPreviewing 2 of 19 pages.\nUpload your study docs or become a member.\nTh3.0\n. If\ni\n←→\nj\n, then\nd\ni\n=\nd\nj\n.\nProof.\nSuppose\np\n(\nm\n)\nij\n>\n0 and\np\n(\nn\n)\nji\n>\n0. Then, by the C-K equation,\np\n(\nm\n+\nn\n)\nii\n=\nX\nk\n∈\nS\np\n(\nm\n)\nik\np\n(\nn\n)\nki\n>\n0\n.\nIt follows that\nm\n+\nn\n=\nkd\ni\nfor some\nk\n≥\n1. Similarly, supposing\np\n(\nl\n)\njj\n>\n0,\nwe have\np\n(\nm\n+\nn\n+\nl\n)\nii\n=\nX\nk\n∈\nS\np\n(\nm\n)\nik\np\n(\nn\n+\nl\n)\nki\n≥\np\n(\nm\n)\nij\np\n(\nl\n)\njj\np\n(\nn\n)\nji\n>\n0\n.\nHence\nm\n+\nn\n+\nl\n=\nk\n1\nd\ni\nfor some\nk\n1\n> k\n. As a consequence, we have\nl\n= (\nk\n1\n-\nk\n)\nd\ni\n=\nc\n1\nd\ni\nwhere\nc\n1\n≥\n1 is an integer.\nRecall that\nd\nj\n=\ngcd\n{\nl\n:\np\n(\nl\n)\njj\n>\n0\n}\n, we have\nd\ni\n≤\nd\nj\n. The same argument shows that\nd\nj\n≤\nd\ni\n, i.e.,\nd\ni\n=\nd\nj\n.\n32\nWant to read all 19 pages?\nPreviewing 3 of 19 pages.\nUpload your study docs or become a member.\nA class is said to be\nperiodic\nif its states are periodic.\nSimilarly, a\nclass\nis said to be aperiodic if its states are aperiodic. Finally, a\nMarkov\nchain\nis said to be\naperiodic\nif all of its states are aperiodic.\nWhy is periodicity important?\nAs we will see later, it plays a role when we discuss limiting distributions.\nIt turns out that in a typical problem, we are given an irreducible Markov\nchain, and we need to check if it is aperiodic.\nHow do we check that a Markov chain is aperiodic?\nConsider a\nfinite irreducible Markov chain\nX\nn\n, i.e., the MC only has a\none class with finite states\n:\n•\nIf there is a self-transition in the chain (\np\nii\n>\n0 for some\ni\n), then the\nchain is aperiodic.\nWant to read all 19 pages?\nPreviewing 4 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 5 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 6 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 7 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 8 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 9 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 10 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 11 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 12 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 13 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 14 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 15 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 16 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 17 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 18 of 19 pages.\nUpload your study docs or become a member.\nWant to read all 19 pages?\nPreviewing 19 of 19 pages.\nUpload your study docs or become a member.\nEnd of preview\nWant to read all 19 pages?\nUpload your study docs or become a member.","attrs_markdown":"[![Course Sidekick logo](https:\/\/assets.coursehero.com\/ssi\/51257f3da318b313edde.svg)](https:\/\/www.coursesidekick.com\/ \"coursesidekick\")\n\n[Documents](https:\/\/www.coursesidekick.com\/sitemap\/documents)\n\n[Q\\&A](https:\/\/www.coursesidekick.com\/sitemap\/questions)\n\nSubjects\n\nStudy Resources\n\nAsk AI\n\nEarn free access\n\nLog in\n\nJoin\n\n# Markov Chains: Periodicity, Recurrence, and Transience Explained\nHelpful\n\nUnhelpful\n\nSchool\n\n[The University of Sydney](https:\/\/www.coursesidekick.com\/schools\/the-university-of-sydney-42)\\*\\*We aren't endorsed by this school\n\nCourse\n\nSTAT 3021\n\nSubject\n\nStatistics\n\nDate\n\nMar 25, 2023\n\nPages\n\n19\n\nUploaded by faisal786khan\n\nDownload\n\nView full document\n\nView full document\n\nHelpful\n\nUnhelpful\n\nDownload\n\n1. [Home](https:\/\/www.coursesidekick.com\/)\/\n2. [Statistics](https:\/\/www.coursesidekick.com\/statistics)\n\nStudents also studied\n\n[![Assignment 1 Recoding Data (Spring 2020)-2.docx](https:\/\/www.coursesidekick.com\/doc-asset\/thumb\/56605ec9cdcdf7bfbc7baeb3c59378e073f6f41e_180.jpg) Assignment 1 Recoding Data (Spring 2020)-2.docx 6 pages Recoding Data Data Assignment 1 Complete the following mini-projects using the 2016 GSS data. I. Depression in the Past Week Suppose we wanted to understand more about the frequency of depression. The GSS asks respondents how often they have felt depresse 4\/20\/2023](https:\/\/www.coursesidekick.com\/statistics\/69155)\n\n[![STAT2215THQ8F22 (1).pdf](https:\/\/www.coursesidekick.com\/doc-asset\/thumb\/3dfed0ea5a79a63d37f4715f146ca5445db14805_180.jpg) STAT2215THQ8F22 (1).pdf 3 pages UNIVERSITY OF CONNECTICUT Statistics Department STAT2215 Take Home Quiz \\#8 Fall 2022 Amir A. Kouzehkanani YOU MUST SHOW ALL NECESSARY WORK Name (print): Victor Han Lec. Sec. \\#: 002 An economist wanted to analyze the relationship between the speed of a car 4\/21\/2023](https:\/\/www.coursesidekick.com\/statistics\/69292)\n\n[![Descriptive Statistics.ppt](https:\/\/www.coursesidekick.com\/doc-asset\/thumb\/e0f8dd491656b988776831216ac2ffdceef9d75a_180.jpg) Descriptive Statistics.ppt 19 pages Populations vs. Samples • We want to describe both samples and populations. • The latter is a matter of inference. \"Outliers\" • Minority cases, so different from the majority that they merit separate consideration - Are they errors? - Are they indicative 4\/21\/2023](https:\/\/www.coursesidekick.com\/statistics\/69300)\n\n[![SolutionPrac2.pdf](https:\/\/www.coursesidekick.com\/doc-asset\/thumb\/55043927d450be65d95d08adfcb775009469b9b0_180.jpg) SolutionPrac2.pdf 2 pages STAT651 Solution to Practice Exam 2 Spring 2023 1. Multiple Choice Questions A. 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By comparing the function plots a 4\/21\/2023](https:\/\/www.coursesidekick.com\/statistics\/69430)\n\nWeek 3 Markov chains:Periodicity, recurrence and transience, positive and null recurrences 7Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have p(n) 00 \\=P(Xn\\= 0\\|X0 \\= 0)6 \\= 0 (= 1),ifn\\= 3,6,9,· · ·, p(n) 00 \\=P(Xn\\= 0\\|X0 \\= 0) = 0,ifn6 \\= 3,6,9,· · ·. Such a state is called aperiodicstate with periodd0\\= 3.\n\nTheperiodof a statei, denoted bydi , is defined as the greatest common divisor (gcd) of alln≥1 for whichp(n) ii \\>0, i.e., di \\= gcd{n:p(n) ii \\>0}. Ifp(n) ii \\= 0, for alln≥1, then we letdi\\= 0 (ordi \\=∞). •Ifdi \\>1, we say that stateiisperiodic. In this case,p(kdi ) ii 6 \\= 0 for allk≥k0, wherek0 \\>1 is an integer, and p(n)ii\\= 0 whenn6 \\=kdi . Note that it happensp(di ) ii \\= 0. •Ifdi \\= 1, we say that stateiisaperiodic. In mathematics, the greatest common divisor (gcd) of two or more in- tegers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. 31\n\n## Want to read all 19 pages?\n**Previewing 2 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\nTh3.0. Ifi←→j, thendi\\=dj . Proof.Supposep(m)ij\\>0 andp(n) ji \\>0\\. Then, by the C-K equation, p(m\\+n)ii \\=X k∈S p(m)ikp(n) ki \\>0. It follows thatm\\+n\\=kdi for somek≥1\\. Similarly, supposingp(l) jj \\>0, we have p(m\\+n\\+l)ii\\=X k∈S p(m)ikp (n\\+l) ki ≥p(m)ijp(l)jjp(n) ji \\>0. Hencem\\+n\\+l\\=k1difor somek1 \\> k. As a consequence, we have l\\= (k1\\-k)di\\=c1d i wherec1≥1 is an integer.Recall thatdj \\=gcd{l:p(l) jj \\>0}, we have di≤dj. The same argument shows thatdj≤di, i.e.,di\\=dj . 32\n\n## Want to read all 19 pages?\n**Previewing 3 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\nA class is said to beperiodicif its states are periodic.Similarly, a classis said to be aperiodic if its states are aperiodic. Finally, aMarkov chainis said to beaperiodicif all of its states are aperiodic. Why is periodicity important? As we will see later, it plays a role when we discuss limiting distributions. It turns out that in a typical problem, we are given an irreducible Markov chain, and we need to check if it is aperiodic. How do we check that a Markov chain is aperiodic? Consider afinite irreducible Markov chainXn , i.e., the MC only has a one class with finite states: •If there is a self-transition in the chain (pii \\>0 for somei), then the chain is aperiodic.\n\n## Want to read all 19 pages?\n**Previewing 4 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 pages?\n**Previewing 5 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 pages?\n**Previewing 6 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 pages?\n**Previewing 7 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 pages?\n**Previewing 8 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 pages?\n**Previewing 9 of 19 pages.** Upload your study docs or become a member.\n\nView full document\n\n## Want to read all 19 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[Facebook](https:\/\/www.facebook.com\/TryCourseSidekick)\n- [Instagram](https:\/\/www.instagram.com\/coursesidekick\/)\n- [YouTube](https:\/\/www.youtube.com\/@CourseSidekick)\n- [Twitter](https:\/\/twitter.com\/Course_Sidekick)\n\n[![Course Sidekick Footer logo](https:\/\/assets.coursehero.com\/ssi\/b26f6b08123071a64384.svg)](https:\/\/www.coursesidekick.com\/ \"Course Sidekick Footer\")\n\nCourse Sidekick, a Learneo, Inc. business\n\n© Learneo, Inc. 2026\n\nCourse Sidekick is not sponsored or endorsed by any college or university.","attrs_readable_markdown":"Week 3 Markov chains:Periodicity, recurrence and transience, positive and null recurrences 7Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have p(n) 00 \\=P(Xn\\= 0\\|X0 \\= 0)6 \\= 0 (= 1),ifn\\= 3,6,9,· · ·, p(n) 00 \\=P(Xn\\= 0\\|X0 \\= 0) = 0,ifn6 \\= 3,6,9,· · ·. Such a state is called aperiodicstate with periodd0\\= 3.\n\nTheperiodof a statei, denoted bydi , is defined as the greatest common divisor (gcd) of alln≥1 for whichp(n) ii \\>0, i.e., di \\= gcd{n:p(n) ii \\>0}. Ifp(n) ii \\= 0, for alln≥1, then we letdi\\= 0 (ordi \\=∞). •Ifdi \\>1, we say that stateiisperiodic. In this case,p(kdi ) ii 6 \\= 0 for allk≥k0, wherek0 \\>1 is an integer, and p(n)ii\\= 0 whenn6 \\=kdi . Note that it happensp(di ) ii \\= 0. •Ifdi \\= 1, we say that stateiisaperiodic. In mathematics, the greatest common divisor (gcd) of two or more in- tegers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. 31\n\n## Want to read all 19 pages?\n**Previewing 2 of 19 pages.** Upload your study docs or become a member.\n\nTh3.0. Ifi←→j, thendi\\=dj . Proof.Supposep(m)ij\\>0 andp(n) ji \\>0\\. Then, by the C-K equation, p(m\\+n)ii \\=X k∈S p(m)ikp(n) ki \\>0. It follows thatm\\+n\\=kdi for somek≥1\\. Similarly, supposingp(l) jj \\>0, we have p(m\\+n\\+l)ii\\=X k∈S p(m)ikp (n\\+l) ki ≥p(m)ijp(l)jjp(n) ji \\>0. Hencem\\+n\\+l\\=k1difor somek1 \\> k. As a consequence, we have l\\= (k1\\-k)di\\=c1d i wherec1≥1 is an integer.Recall thatdj \\=gcd{l:p(l) jj \\>0}, we have di≤dj. The same argument shows thatdj≤di, i.e.,di\\=dj . 32\n\n## Want to read all 19 pages?\n**Previewing 3 of 19 pages.** Upload your study docs or become a member.\n\nA class is said to beperiodicif its states are periodic.Similarly, a classis said to be aperiodic if its states are aperiodic. Finally, aMarkov chainis said to beaperiodicif all of its states are aperiodic. Why is periodicity important? As we will see later, it plays a role when we discuss limiting distributions. It turns out that in a typical problem, we are given an irreducible Markov chain, and we need to check if it is aperiodic. How do we check that a Markov chain is aperiodic? Consider afinite irreducible Markov chainXn , i.e., the MC only has a one class with finite states: •If there is a self-transition in the chain (pii \\>0 for somei), then the chain is aperiodic.\n\n## Want to read all 19 pages?\n**Previewing 4 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 5 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 6 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 7 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 8 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 9 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 10 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 11 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 12 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 13 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 14 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 15 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 16 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 17 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 18 of 19 pages.** Upload your study docs or become a member.\n\n## Want to read all 19 pages?\n**Previewing 19 of 19 pages.** Upload your study docs or become a member.\n\n## End of preview\n**Want to read all 19 pages?** Upload your study docs or become a member.","meta_canonical":null}

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Meta Description	Statistics document from The University of Sydney, 19 pages, Week 3 Markov chains: Periodicity, recurrence and transience, positive and null recurrences 7 Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have (n) p00 = P (Xn = 0 \| X0 = 0) 6= 0 (=
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Boilerpipe Text	Week 3 Markov chains: Periodicity, recurrence and transience, positive and null recurrences 7 Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have p ( n ) 00 = P ( X n = 0 \| X 0 = 0) 6 = 0 (= 1) , if n = 3 , 6 , 9 , · · · , p ( n ) 00 = P ( X n = 0 \| X 0 = 0) = 0 , if n 6 = 3 , 6 , 9 , · · · . Such a state is called a periodic state with period d 0 = 3. The period of a state i , denoted by d i , is defined as the greatest common divisor (gcd) of all n ≥ 1 for which p ( n ) ii > 0, i.e., d i = gcd { n : p ( n ) ii > 0 } . If p ( n ) ii = 0, for all n ≥ 1, then we let d i = 0 (or d i = ∞ ). • If d i > 1, we say that state i is periodic . In this case, p ( kd i ) ii 6 = 0 for all k ≥ k 0 , where k 0 > 1 is an integer, and p ( n ) ii = 0 when n 6 = kd i . Note that it happens p ( d i ) ii = 0. • If d i = 1, we say that state i is aperiodic . In mathematics, the greatest common divisor (gcd) of two or more in- tegers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. 31 Want to read all 19 pages? Previewing 2 of 19 pages. Upload your study docs or become a member. Th3.0 . If i ←→ j , then d i = d j . Proof. Suppose p ( m ) ij > 0 and p ( n ) ji > 0. Then, by the C-K equation, p ( m + n ) ii = X k ∈ S p ( m ) ik p ( n ) ki > 0 . It follows that m + n = kd i for some k ≥ 1. Similarly, supposing p ( l ) jj > 0, we have p ( m + n + l ) ii = X k ∈ S p ( m ) ik p ( n + l ) ki ≥ p ( m ) ij p ( l ) jj p ( n ) ji > 0 . Hence m + n + l = k 1 d i for some k 1 > k . As a consequence, we have l = ( k 1 - k ) d i = c 1 d i where c 1 ≥ 1 is an integer. Recall that d j = gcd { l : p ( l ) jj > 0 } , we have d i ≤ d j . The same argument shows that d j ≤ d i , i.e., d i = d j . 32 Want to read all 19 pages? Previewing 3 of 19 pages. Upload your study docs or become a member. A class is said to be periodic if its states are periodic. Similarly, a class is said to be aperiodic if its states are aperiodic. Finally, a Markov chain is said to be aperiodic if all of its states are aperiodic. Why is periodicity important? As we will see later, it plays a role when we discuss limiting distributions. It turns out that in a typical problem, we are given an irreducible Markov chain, and we need to check if it is aperiodic. How do we check that a Markov chain is aperiodic? Consider a finite irreducible Markov chain X n , i.e., the MC only has a one class with finite states : • If there is a self-transition in the chain ( p ii > 0 for some i ), then the chain is aperiodic. Want to read all 19 pages? Previewing 4 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 5 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 6 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 7 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 8 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 9 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 10 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 11 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 12 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 13 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 14 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 15 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 16 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 17 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 18 of 19 pages. Upload your study docs or become a member. Want to read all 19 pages? Previewing 19 of 19 pages. Upload your study docs or become a member. End of preview Want to read all 19 pages? Upload your study docs or become a member.
Markdown	[![Course Sidekick logo](https://assets.coursehero.com/ssi/51257f3da318b313edde.svg)](https://www.coursesidekick.com/ "coursesidekick") [Documents](https://www.coursesidekick.com/sitemap/documents) [Q\&A](https://www.coursesidekick.com/sitemap/questions) Subjects Study Resources Ask AI Earn free access Log in Join # Markov Chains: Periodicity, Recurrence, and Transience Explained Helpful Unhelpful School [The University of Sydney](https://www.coursesidekick.com/schools/the-university-of-sydney-42)\\We aren't endorsed by this school Course STAT 3021 Subject Statistics Date Mar 25, 2023 Pages 19 Uploaded by faisal786khan Download View full document View full document Helpful Unhelpful Download 1. [Home](https://www.coursesidekick.com/)/ 2. [Statistics](https://www.coursesidekick.com/statistics) Students also studied [![Assignment 1 Recoding Data (Spring 2020)-2.docx](https://www.coursesidekick.com/doc-asset/thumb/56605ec9cdcdf7bfbc7baeb3c59378e073f6f41e_180.jpg) Assignment 1 Recoding Data (Spring 2020)-2.docx 6 pages Recoding Data Data Assignment 1 Complete the following mini-projects using the 2016 GSS data. I. Depression in the Past Week Suppose we wanted to understand more about the frequency of depression. The GSS asks respondents how often they have felt depresse 4/20/2023](https://www.coursesidekick.com/statistics/69155) [![STAT2215THQ8F22 (1).pdf](https://www.coursesidekick.com/doc-asset/thumb/3dfed0ea5a79a63d37f4715f146ca5445db14805_180.jpg) STAT2215THQ8F22 (1).pdf 3 pages UNIVERSITY OF CONNECTICUT Statistics Department STAT2215 Take Home Quiz \#8 Fall 2022 Amir A. Kouzehkanani YOU MUST SHOW ALL NECESSARY WORK Name (print): Victor Han Lec. Sec. \#: 002 An economist wanted to analyze the relationship between the speed of a car 4/21/2023](https://www.coursesidekick.com/statistics/69292) [![Descriptive Statistics.ppt](https://www.coursesidekick.com/doc-asset/thumb/e0f8dd491656b988776831216ac2ffdceef9d75a_180.jpg) Descriptive Statistics.ppt 19 pages Populations vs. Samples • We want to describe both samples and populations. • The latter is a matter of inference. "Outliers" • Minority cases, so different from the majority that they merit separate consideration - Are they errors? - Are they indicative 4/21/2023](https://www.coursesidekick.com/statistics/69300) [![SolutionPrac2.pdf](https://www.coursesidekick.com/doc-asset/thumb/55043927d450be65d95d08adfcb775009469b9b0_180.jpg) SolutionPrac2.pdf 2 pages STAT651 Solution to Practice Exam 2 Spring 2023 1. Multiple Choice Questions A. (c) The p-value \< 0.05, but we do not know if it is less or greater than 0.01. B. 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(e) None of the above. 4/21/2023](https://www.coursesidekick.com/statistics/69405) [![Practice Quiz Unit 12.pdf](https://www.coursesidekick.com/doc-asset/thumb/386063b8be9191e8e6bb263aeea923f525ec4e11_180.jpg) Practice Quiz Unit 12.pdf 11 pages Practice Quiz Chapter 12 Marisell Fournier (username: 900891685) Attempt 1 Written: Mar 19, 2023 2:42 PM - Mar 19, 2023 4:08 PM Submission View Released: Oct 17, 2018 3:57 PM 1 / 1 point If we wish to determine whether there is evidence that the proportio 4/21/2023](https://www.coursesidekick.com/statistics/69415) [![Radioactive Lab PHY224 Conclusion.pdf](https://www.coursesidekick.com/doc-asset/thumb/7603a4859db9777d9fdeafa932cb7c5c7690876b_180.jpg) Radioactive Lab PHY224 Conclusion.pdf 1 pages 1 Part 1: Nonlinear Fitting Model Conclusion In this exercise, we are finding whether a linear or nonlinear model is a good trend predictor and whether both prediction model matches the theoretical half-life of Barium137. By comparing the function plots a 4/21/2023](https://www.coursesidekick.com/statistics/69430) Week 3 Markov chains:Periodicity, recurrence and transience, positive and null recurrences 7Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have p(n) 00 \=P(Xn\= 0\\|X0 \= 0)6 \= 0 (= 1),ifn\= 3,6,9,· · ·, p(n) 00 \=P(Xn\= 0\\|X0 \= 0) = 0,ifn6 \= 3,6,9,· · ·. Such a state is called aperiodicstate with periodd0\= 3. Theperiodof a statei, denoted bydi , is defined as the greatest common divisor (gcd) of alln≥1 for whichp(n) ii \>0, i.e., di \= gcd{n:p(n) ii \>0}. Ifp(n) ii \= 0, for alln≥1, then we letdi\= 0 (ordi \=∞). •Ifdi \>1, we say that stateiisperiodic. In this case,p(kdi ) ii 6 \= 0 for allk≥k0, wherek0 \>1 is an integer, and p(n)ii\= 0 whenn6 \=kdi . Note that it happensp(di ) ii \= 0. •Ifdi \= 1, we say that stateiisaperiodic. In mathematics, the greatest common divisor (gcd) of two or more in- tegers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. 31 ## Want to read all 19 pages? Previewing 2 of 19 pages. Upload your study docs or become a member. View full document Th3.0. Ifi←→j, thendi\=dj . Proof.Supposep(m)ij\>0 andp(n) ji \>0\. Then, by the C-K equation, p(m\+n)ii \=X k∈S p(m)ikp(n) ki \>0. It follows thatm\+n\=kdi for somek≥1\. Similarly, supposingp(l) jj \>0, we have p(m\+n\+l)ii\=X k∈S p(m)ikp (n\+l) ki ≥p(m)ijp(l)jjp(n) ji \>0. Hencem\+n\+l\=k1difor somek1 \> k. As a consequence, we have l\= (k1\-k)di\=c1d i wherec1≥1 is an integer.Recall thatdj \=gcd{l:p(l) jj \>0}, we have di≤dj. The same argument shows thatdj≤di, i.e.,di\=dj . 32 ## Want to read all 19 pages? Previewing 3 of 19 pages. Upload your study docs or become a member. View full document A class is said to beperiodicif its states are periodic.Similarly, a classis said to be aperiodic if its states are aperiodic. Finally, aMarkov chainis said to beaperiodicif all of its states are aperiodic. Why is periodicity important? As we will see later, it plays a role when we discuss limiting distributions. It turns out that in a typical problem, we are given an irreducible Markov chain, and we need to check if it is aperiodic. How do we check that a Markov chain is aperiodic? Consider afinite irreducible Markov chainXn , i.e., the MC only has a one class with finite states: •If there is a self-transition in the chain (pii \>0 for somei), then the chain is aperiodic. ## Want to read all 19 pages? Previewing 4 of 19 pages. Upload your study docs or become a member. View full document ## Want to read all 19 pages? Previewing 5 of 19 pages. Upload your study docs or become a member. View full document ## Want to read all 19 pages? Previewing 6 of 19 pages. 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Readable Markdown	Week 3 Markov chains:Periodicity, recurrence and transience, positive and null recurrences 7Markov chain: periodicity Consider the Markov chain shown below: There is a periodic pattern in this chain. Indeed, we have p(n) 00 \=P(Xn\= 0\\|X0 \= 0)6 \= 0 (= 1),ifn\= 3,6,9,· · ·, p(n) 00 \=P(Xn\= 0\\|X0 \= 0) = 0,ifn6 \= 3,6,9,· · ·. Such a state is called aperiodicstate with periodd0\= 3. Theperiodof a statei, denoted bydi , is defined as the greatest common divisor (gcd) of alln≥1 for whichp(n) ii \>0, i.e., di \= gcd{n:p(n) ii \>0}. Ifp(n) ii \= 0, for alln≥1, then we letdi\= 0 (ordi \=∞). •Ifdi \>1, we say that stateiisperiodic. In this case,p(kdi ) ii 6 \= 0 for allk≥k0, wherek0 \>1 is an integer, and p(n)ii\= 0 whenn6 \=kdi . Note that it happensp(di ) ii \= 0. •Ifdi \= 1, we say that stateiisaperiodic. In mathematics, the greatest common divisor (gcd) of two or more in- tegers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. 31 ## Want to read all 19 pages? Previewing 2 of 19 pages. Upload your study docs or become a member. Th3.0. Ifi←→j, thendi\=dj . Proof.Supposep(m)ij\>0 andp(n) ji \>0\. Then, by the C-K equation, p(m\+n)ii \=X k∈S p(m)ikp(n) ki \>0. It follows thatm\+n\=kdi for somek≥1\. Similarly, supposingp(l) jj \>0, we have p(m\+n\+l)ii\=X k∈S p(m)ikp (n\+l) ki ≥p(m)ijp(l)jjp(n) ji \>0. Hencem\+n\+l\=k1difor somek1 \> k. As a consequence, we have l\= (k1\-k)di\=c1d i wherec1≥1 is an integer.Recall thatdj \=gcd{l:p(l) jj \>0}, we have di≤dj. The same argument shows thatdj≤di, i.e.,di\=dj . 32 ## Want to read all 19 pages? Previewing 3 of 19 pages. Upload your study docs or become a member. A class is said to beperiodicif its states are periodic.Similarly, a classis said to be aperiodic if its states are aperiodic. Finally, aMarkov chainis said to beaperiodicif all of its states are aperiodic. Why is periodicity important? As we will see later, it plays a role when we discuss limiting distributions. It turns out that in a typical problem, we are given an irreducible Markov chain, and we need to check if it is aperiodic. How do we check that a Markov chain is aperiodic? Consider afinite irreducible Markov chainXn , i.e., the MC only has a one class with finite states: •If there is a self-transition in the chain (pii \>0 for somei), then the chain is aperiodic. ## Want to read all 19 pages? Previewing 4 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 5 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 6 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 7 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 8 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 9 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 10 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 11 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 12 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 13 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 14 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 15 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 16 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 17 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 18 of 19 pages. Upload your study docs or become a member. ## Want to read all 19 pages? Previewing 19 of 19 pages. Upload your study docs or become a member. ## End of preview Want to read all 19 pages? Upload your study docs or become a member.
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