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Asymptotics for the moments of the overshoot and undershoot of a random walk

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Zhaolei Cui ,
Yuebao Wang  and
Kaiyong Wang
Zhaolei Cui*
Affiliation:
Soochow University
Yuebao Wang*
Affiliation:
Soochow University
Kaiyong Wang*
Affiliation:
Suzhou University of Science and Technology
*
∗ Postal address: Department of Mathematics, Soochow University, Suzhou, 215006, P. R. China.
∗ Postal address: Department of Mathematics, Soochow University, Suzhou, 215006, P. R. China.
∗∗∗ Postal address: Department of Information and Computational Science, School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, P. R. China.
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Abstract

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In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

Keywords

Local and nonlocal asymptoticsmomentfirst ascending ladder heightrenewal equationovershootundershoot

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K30: Applications (congestion, allocation, storage, traffic, etc.)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 469 - 494
Copyright
Copyright © Applied Probability Trust 2009 

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