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Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Cheng-Der Fuh  and
Tze Leung Lai
Cheng-Der Fuh*
Affiliation:
Academia Sinica
Tze Leung Lai*
Affiliation:
Stanford University
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC.
∗∗ Postal address: Department of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305-4065, USA. Email address: lait@stat.stanford.edu
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Abstract

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (X n ,S n ), n ≥0, in which X n takes values in a general state space and S n takes values in ℝd . In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when S n exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for S n . For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (X T ,S T ), where T = inf { n : S n,1 > b } and S n,1 denotes the first component of S n .

Keywords

First passage timeMarkov random walkMarkov renewal theoryrate of convergence

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K15: Markov renewal processes, semi-Markov processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 652 - 673
Copyright
Copyright © Applied Probability Trust 2001 

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