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Diskr. Mat., 2014, Volume 26, Issue 3, Pages 79–89 (Mi dm1292)  

This article is cited in 7 scientific papers (total in 7 papers)

On repetitions of long tuples in a Markov chain

V. G. Mikhailova, A. M. Shoitovb

a Steklov Mathematical Institute of Russian Academy of Sciences
b Academy of Criptography of Russia

Abstract: Let $X_0,X_1,\ldots$ be a simple ergodic finite Markov chain. We prove limit theorems for the distribution of the number $\tilde\xi(s,n)$ of events
$$\{X_{i-1}\ne X_{j-1}, X_{i+k}= X_{j+k}, k=0,\ldots,s-1\},\quad 1\le i<j\le n,$$
when $s,n\to\infty$. Limit theorems for distributions of some random variables connected with $\tilde\xi(s,n)$ are derived as corollaries.

Keywords: Markov chain, repetitions of tuples, Poisson limit theorem.

DOI: https://doi.org/10.4213/dm1292

Full text: PDF file (475 kB)
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English version:
Discrete Mathematics and Applications, 2015, 25:5, 295–303

Bibliographic databases:

Document Type: Article
UDC: 519.212.2+519.214
Received: 17.02.2014

Citation: V. G. Mikhailov, A. M. Shoitov, “On repetitions of long tuples in a Markov chain”, Diskr. Mat., 26:3 (2014), 79–89; Discrete Math. Appl., 25:5 (2015), 295–303

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Vladimir G. Mikhaylov, “Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain”, Discrete Math. Appl., 26:2 (2016), 105–113  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. V. G. Mikhailov, A. M. Shoitov, “Mnogokratnye povtoreniya dlinnykh tsepochek v konechnoi tsepi Markova”, Matem. vopr. kriptogr., 6:3 (2015), 117–133  mathnet  crossref  mathscinet  elib
    3. N. M. Mezhennaya, “O chisle sovpadenii znakov v diskretnoi sluchainoi posledovatelnosti, upravlyaemoi tsepyu Markova”, Sib. elektron. matem. izv., 13 (2016), 305–317  mathnet  crossref
    4. V. G. Mikhailov, “On the probability of existence of substrings with the same structure in a random sequence”, Discrete Math. Appl., 27:6 (2017), 377–386  mathnet  crossref  crossref  mathscinet  isi  elib
    5. N. M. Mezhennaya, “Otsenka dlya raspredeleniya chisel serii v sluchainoi posledovatelnosti, upravlyaemoi statsionarnoi tsepyu Markova”, PDM, 2017, no. 35, 14–28  mathnet  crossref
    6. V. G. Mikhailov, “On the reduction property of the number of $H$-equivalent tuples of states in a discrete Markov chain”, Discrete Math. Appl., 28:2 (2018), 75–82  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. V. A. Voloshko, Yu. S. Kharin, “Semibinomialnye uslovno nelineinye avtoregressionnye modeli diskretnykh sluchainykh posledovatelnostei: veroyatnostnye svoistva i statisticheskoe otsenivanie parametrov”, Diskret. matem., 31:1 (2019), 72–98  mathnet  crossref  elib
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