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 Teor. Veroyatnost. i Primenen., 1958, Volume 3, Issue 4, Pages 361–385 (Mi tvp4943)

Limit Theorems for Markov Chains with a Finite Number of States

L. D. Meshalkin

Moscow

Abstract: Consider the scheme of trial sequences
$$\nu _{11}\nu_{21},\nu_{22} \nu_{n1},\nu_{n2},…,\nu_{nn} ………$$
The sequence $\nu_{nk}$, $k=1,…,n$, is a uniform Markov chain with a finite number of states $E_1,…,E_s$ and a given matrix of transition probabilities
$$P=P(n)=\|{p_{uv}(n)}\|_{u,v=1}^s.$$

Let $\mu=\mu (n)$ denote the number of passages up in the $n$-th sequence of trials of the system through $E_1$ on condition that the system is in state $E_1$ at the initial (or zero-th) time. We consider the limit distribution for a sequence of random variables
$$\alpha(\mu-n\theta),\quad\alpha=\alpha(n),\quad\theta=\theta(n).$$

Theorems 1–5 give characteristic functions for some possible limit distributions.
The main result of this paper is Theorem 6:
If the limit distribution for $\alpha(\mu-n\theta)$ exists, then it does not differ from one of those found in Theorems 1–5 by more than a linear transformation.

Full text: PDF file (2296 kB)

English version:
Theory of Probability and its Applications, 1958, 3:4, 335–357

Citation: L. D. Meshalkin, “Limit Theorems for Markov Chains with a Finite Number of States”, Teor. Veroyatnost. i Primenen., 3:4 (1958), 361–385; Theory Probab. Appl., 3:4 (1958), 335–357

Citation in format AMSBIB
\Bibitem{Mes58} \by L.~D.~Meshalkin \paper Limit Theorems for Markov Chains with a Finite Number of States \jour Teor. Veroyatnost. i Primenen. \yr 1958 \vol 3 \issue 4 \pages 361--385 \mathnet{http://mi.mathnet.ru/tvp4943} \transl \jour Theory Probab. Appl. \yr 1958 \vol 3 \issue 4 \pages 335--357 \crossref{https://doi.org/10.1137/1103029} 

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Erratum

This publication is cited in the following articles:
1. I. A. Ibragimov, “Dobrushin's works on Markov processes”, Russian Math. Surveys, 52:2 (1997), 239–243
2. S. A. Aivazyan, L. G. Afanas'eva, V. M. Buchstaber, Yu. N. Blagoveshchenskii, B. M. Gurevich, Yu. V. Prokhorov, Ya. G. Sinai, V. M. Tikhomirov, A. N. Shiryaev, “Lev Dmitrievich Meshalkin (obituary)”, Russian Math. Surveys, 56:3 (2001), 563–568
3. Silvestrov D. Silvestrov S., “Asymptotic Expansions For Stationary Distributions of Perturbed Semi-Markov Processes”, Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures For Networks, Data Classification and Optimization, Springer Proceedings in Mathematics & Statistics, 179, ed. Silvestrov S. Rancic M., Springer International Publishing Ag, 2016, 151–222