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

This article is cited in 10 scientific papers (total in 10 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:

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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\paper On repetitions of long tuples in a Markov chain
\jour Diskr. Mat.
\yr 2014
\vol 26
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\pages 79--89
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\jour Discrete Math. Appl.
\yr 2015
\vol 25
\issue 5
\pages 295--303
\crossref{https://doi.org/10.1515/dma-2015-0028}
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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, “Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation”, Discrete Math. Appl., 30:6 (2020), 417–437  mathnet  crossref  crossref  mathscinet  isi  elib
    8. N. M. Mezhennaya, “O chisle $f$-rekurrentnykh serii i tsepochek v konechnoi tsepi Markova”, PDM. Prilozhenie, 2019, no. 12, 18–21  mathnet  crossref  elib
    9. V. G. Mikhailov, N. M. Mezhennaya, “Normal approximation for $U$- and $V$-statistics of a stationary absolutely regular sequence”, Sib. elektron. matem. izv., 17 (2020), 672–682  mathnet  crossref
    10. V. G. Mikhailov, N. M. Mezhennaya, A. V. Volgin, “Ob usloviyakh asimptoticheskoi normalnosti chisla povtorenii v statsionarnoi sluchainoi posledovatelnosti”, Diskret. matem., 33:3 (2021), 64–78  mathnet  crossref
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