This article is cited 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
Fulltext:
PDF file (2296 kB)
