Limit Theorems for Markov Chains with a Finite Number of States

L. D. Meshalkin

Teoriâ veroâtnostej i ee primeneniâ, Tome 3 (1958) no. 4, pp. 361-385

Résumé

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.