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On the length of the longest run in a multi-state Markov chain

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  • Vaggelatou, Eutichia

Abstract

Let {Xa}a[set membership, variant]Z be an irreducible and aperiodic Markov chain on a finite state space S={0,1,...,r}, r[greater-or-equal, slanted]1. Denote by Ln the length of the longest run of consecutive i's, for i=1,...,r, that occurs in the sequence X1,...,Xn. In this work, we extend a result of Goncharov (Amer. Math. Soc. Transl. 19 (1943) 1) which concerned a limit law for Ln in sequences of 0-1 i.i.d. trials. Moreover, it is shown that Ln has approximately an extreme value distribution along a certain subsequence. Finally, a weak version of an Erdös-Rényi type law for Ln is proved.

Suggested Citation

  • Vaggelatou, Eutichia, 2003. "On the length of the longest run in a multi-state Markov chain," Statistics & Probability Letters, Elsevier, vol. 62(3), pages 211-221, April.
    • Handle: RePEc:eee:stapro:v:62:y:2003:i:3:p:211-221
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    Citations

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    Cited by:

    1. Serkan Eryilmaz, 2005. "On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials," Statistical Papers, Springer, vol. 46(1), pages 117-128, January.
    2. George Haiman, 2012. "1-Dependent Stationary Sequences for Some Given Joint Distributions of Two Consecutive Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 445-458, September.
    3. Novak, S.Y., 2017. "On the length of the longest head run," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 111-114.
    4. Ghosh, Souvik & Samorodnitsky, Gennady, 2010. "Long strange segments, ruin probabilities and the effect of memory on moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2302-2330, December.
    5. EryIlmaz, Serkan, 2008. "Distribution of runs in a sequence of exchangeable multi-state trials," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1505-1513, September.

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