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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


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.


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

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

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


Vol.2 • No. 3 • June, 1974
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