Limit Theorems for Markov Chains with a Finite Number of States

Abstract

Consider the scheme of trial sequences \[ \begin{gathered} \nu _{11} \hfill \\ \nu _{21} ,\nu _{22} \hfill \\ \cdots \hfill \\ \nu _{n1} ,\nu _{n2} , \cdots ,\nu _{nn} \hfill \\ \cdots \cdots \cdots \hfill \\ \end{gathered} \] The sequence $\nu _{nk} $, $k = 1, \cdots ,n$, is a uniform Markov chain with a finite number of states $E_1 , \cdots ,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 ound in Theorems 1–5 by more than a linear transformation.