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Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Cheng-Der Fuh
Cheng-Der Fuh*
Affiliation:
National Central University and Academia Sinica
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC. Email address: stcheng@stat.sinica.edu.tw
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Abstract

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Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Keywords

Boundary crossing probabilityladder height distributionMarkov-dependent Wald martingaleovershootPoisson equationuniform Markov renewal theory

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
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 826 - 852
Copyright
Copyright © Applied Probability Trust 2007 

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