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Diffusion approximations for the maximum of a perturbed random walk

Part of: Markov processes Limit theorems Operations research and management science Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Victor F. Araman  and
Peter W. Glynn
Victor F. Araman*
Affiliation:
New York University
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Stern School of Business, New York University, New York, NY 10012-1126, USA. Email address: varaman@stern.nyu.edu
∗∗ Postal address: Management Science and Engineering, Stanford University, Stanford, CA 94305-5015, USA. Email address: glynn@stanford.edu
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Abstract

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Consider a random walk S=(Sn: n≥0) that is ‘perturbed’ by a stationary sequence (ξn: n≥0) to produce the process S=(Sn+ξn: n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of Mn=max{Sk+ξk: 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

Keywords

Perturbed random walkdiffusion approximationlight-tailed distributionheavy-tailed distributionlimit theorem

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes 60G99: None of the above, but in this section
Secondary: 60G70: Extreme value theory; extremal processes 90B30: Production models
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 663 - 680
Copyright
Copyright © Applied Probability Trust 2005 

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