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Teoriya Veroyatnostei i ee Primeneniya, 1958, Volume 3, Issue 4, Pages 361–385 (Mi tvp4943)  

This article is cited in 11 scientific papers (total in 12 papers)

Limit Theorems for Markov Chains with a Finite Number of States

L. D. Meshalkin

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

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.
Received: 21.02.1958
English version:
Theory of Probability and its Applications, 1958, Volume 3, Issue 4, Pages 335–357
DOI: https://doi.org/10.1137/1103029
Document Type: Article
Language: Russian
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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  • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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