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Teoriya Veroyatnostei i ee Primeneniya, 1958, Volume 3, Issue 4, Pages 361–385
(Mi tvp4943)
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This article is cited in 10 scientific papers (total in 11 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
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
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