NDLI: Limit Theorems for Markov Chains with a Finite Number of States

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Meshalkin, L. D.
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 15 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 15 by more than a linear transformation.
Starting Page	335
Ending Page	357
Page Count	23
File Format	PDF
ISSN	0040585X
DOI	10.1137/1103029
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	4
Volume Number	3
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-17
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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