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June, 1974 Renewal Theory for Functionals of a Markov Chain with General State Space
Harry Kesten
Ann. Probab. 2(3): 355-386 (June, 1974). DOI: 10.1214/aop/1176996654

Abstract

We prove an analogue of Blackwell's renewal theorem or the "key renewal theorem" and the existence of the limit distribution of the residual waiting time in the following setup: $X_0, X_1, \cdots$ is a Markov chain with separable metric state space and $u_0, u_1, \cdots$ is a sequence of random variables, such that the conditional distribution of $u_i$, given all $X_j$ and $u_l, l \neq i$, depends on $X_i$ and $X_{i+1}$ only. Here the $V_n \equiv \sum^{n-1}_0 u_i, n \geqq 1$, take the role of the partial sums of independent identically distributed random variables in ordinary renewal theory. E.g. the key renewal theorem in this setup states that $\lim_{t\rightarrow\infty} E\{\sum^\infty_{n=0} g(X_n, t - V_n)\mid X_0 = x\}$ exists for suitable $g(\bullet, \bullet)$, and is independent of $x$.

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Harry Kesten. "Renewal Theory for Functionals of a Markov Chain with General State Space." Ann. Probab. 2 (3) 355 - 386, June, 1974. https://doi.org/10.1214/aop/1176996654

Information

Published: June, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0303.60090
MathSciNet: MR365740
Digital Object Identifier: 10.1214/aop/1176996654

Subjects:
Primary: 60K05
Secondary: 60B99 , 60J10 , 60K15

Keywords: Markov chains with general state space , Products of random matrices , Renewal theory semi Markov chains , residual waiting time

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 3 • June, 1974
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