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Discrete Mathematics and Applications

Editor-in-Chief: Zubkov, Andrei

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CiteScore 2016: 0.16

SCImago Journal Rank (SJR) 2016: 0.231
Source Normalized Impact per Paper (SNIP) 2016: 0.552

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1569-3929
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Volume 26, Issue 2

Issues

Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain

Vladimir G. Mikhaylov
Published Online: 2016-04-26 | DOI: https://doi.org/10.1515/dma-2016-0008

Abstract

Let X0, X1, … be a simple ergodic Markov chain with a finite set of states and ξ̃n, k(s) be a number of runs of k-fold repetitions of strings having length s. Estimates of accuracy of the Poisson approximation for the distribution of ξn, k(s) in the sequence X0, X1, …, Xn+s−1 are obtained, these estimates are uniform over k.

Keywords: Markov chain; k-fold repetitions of s-strings; accuracy of the Poisson approximation

Originally published in Diskretnaya Matematika (2015) 27,.4, 67–78 (in Russian)

References

  • [1]

    Belyaev P.F., “On the problem of the joint distribution of frequencies of s-tuples in compound Markov chains with a large number of states”, Theory Probab. Appl., 14: 2 (1969), 324–330.Google Scholar

  • [2]

    Zubkov A.M., Mikhaylov V.G., “Limit distributions of random variables associated with long duplications in a sequence of independent trials”, Theory Probab. Appl., 19: 1 (1974), 172–179.Google Scholar

  • [3]

    Zubkov A.M., Mikhaylov V.G., “Repetitions of s-tuples in a sequence of independent trials”, Theory Probab. Appl., 24: 2 (1980), 269–282.Google Scholar

  • [4]

    Mikhaylov V.G., “Limit distribution of random variables associated with multiple long duplications in a sequence of independent trials”, Theory Probab. Appl., 19: 1 (1974), 180–184.Google Scholar

  • [5]

    Hansen N.R., “Local alignment of Markov chains”, Ann. Appl. Probab., 16: 3 (2006), 1262–1296.Google Scholar

  • [6]

    Belyaev P.F., “On the problem of the joint distribution of frequencies of s-tuples in compound Markov chains with a large number of states”, Theory Probab. Appl., 14: 2 (1969), 324–330.Google Scholar

  • [7]

    Belyaev P.F., “On the joint frequency distribution of outcomes in Markov chains with a large number of states”, Theory Probab. Appl., 22: 3 (1978), 521–532.Google Scholar

  • [8]

    Zubkov A.M., “Inequalities for transition probabilities with taboos and their applications”, Math. USSR-Sb., 37: 4 (1980), 451–588.Google Scholar

  • [9]

    Mikhaylov V.G., Shoytov A.M., “On repetitions of long tuples in a Markov chain”, Discrete Math. Appl., 25: 5 (2015), 295–303.Google Scholar

  • [10]

    Mikhaylov V.G., Shoytov A.M., “On multiple repetitions of long tuples in a Markov chain”, Mat. Vopr. Kriptogr., 6: 3 (2015), 117–133 (in Russian).Google Scholar

  • [11]

    Gantmacher F.R., The Theory of Matrices. vol. 1 and vol. 2, Chelsea Publishing Company, New York, 1959, vol. 1: x+374 pp. vol. 2: x+277 pp. pp.Google Scholar

  • [12]

    Shoytov A.M., “The Poisson approximation for the number of matches of values of a discrete function from chains”, Discrete Math. Appl., 15: 3 (2005), 241–254.Google Scholar

  • [13]

    Barbour A.D., Holst L., Janson S., Poisson Approximation, Oxford University Press, 1992, 277 pp.Google Scholar

  • [14]

    Erhardsson T., “Stein’s method for Poisson and compound Poisson approximation”, In “An introduction to Stein’s method”, eds. Barbour A. D., Chen L. H. Y., Singapore Univ. Press, 2005, 61–113.Google Scholar

About the article

Received: 2015-07-07

Published Online: 2016-04-26

Published in Print: 2016-04-01


Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-50-00005

This work was supported by the Russian Science Foundation under grant no. 14-50-00005


Citation Information: Discrete Mathematics and Applications, Volume 26, Issue 2, Pages 105–113, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265, DOI: https://doi.org/10.1515/dma-2016-0008.

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