Open Access
Translator Disclaimer
August, 1978 On the Range of Recurrent Markov Chains
Leo Chosid, Richard Isaac
Ann. Probab. 6(4): 680-687 (August, 1978). DOI: 10.1214/aop/1176995489

Abstract

Let $R_n$ be the number of distinct elements among $X_0, X_1,\cdots, X_n$, where $\{X_n\}$ is an irreducible recurrent Markov chain. It is shown that, under an appropriate condition, $n^{-1}R_n \rightarrow 0$ a.s. $(P_a)$ where $a$ is any state and $P_a$ is conditional probability measure given $X_0 = a$. We prove that any recurrent random walk satisfies our condition, so that the result contains the well-known random walk case. We also give an example of an irreducible recurrent chain for which the result fails to hold.

Citation

Download Citation

Leo Chosid. Richard Isaac. "On the Range of Recurrent Markov Chains." Ann. Probab. 6 (4) 680 - 687, August, 1978. https://doi.org/10.1214/aop/1176995489

Information

Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0388.60064
MathSciNet: MR474507
Digital Object Identifier: 10.1214/aop/1176995489

Subjects:
Primary: 60J10
Secondary: 60F15

Rights: Copyright © 1978 Institute of Mathematical Statistics

JOURNAL ARTICLE
8 PAGES


SHARE
Vol.6 • No. 4 • August, 1978
Back to Top