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August, 1979 Strong Ratio Limit Theorems for $\phi$-Recurrent Markov Chains
E. Nummelin
Ann. Probab. 7(4): 639-650 (August, 1979). DOI: 10.1214/aop/1176994987

Abstract

Let $\{X_n; n = 0, 1, \cdots\}$ be a $\phi$-recurrent Markov chain on a general measurable state space $(S, \mathscr{F})$ with transition probabilities $P(x, A), x \in S, A \in \mathscr{F}$. The convergence of the ratio $\lambda P^{n+m}f / \mu P^ng$ (as $n \rightarrow \infty$), where $\lambda$ and $\mu$ are nonnegative measures on $(S, \mathscr{F})$ and $f$ and $g$ are nonnegative measurable functions on $S$, is studied. We show that the ratio converges, provided that $\lambda, \mu, f$ and $g$ are in a certain sense "small," and provided that for an embedded renewal sequence $\{u(n)\}$ the limit $\lim u(n + 1)/u(n)$ exists.

Citation

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E. Nummelin. "Strong Ratio Limit Theorems for $\phi$-Recurrent Markov Chains." Ann. Probab. 7 (4) 639 - 650, August, 1979. https://doi.org/10.1214/aop/1176994987

Information

Published: August, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0411.60069
MathSciNet: MR537211
Digital Object Identifier: 10.1214/aop/1176994987

Subjects:
Primary: 60J10

Keywords: $\phi$-recurrent , $R$-recurrent , Markov chain , quasi-stationary distribution , strong ratio limit

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 4 • August, 1979
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