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Series expansions for the all-time maximum of α-stable random walks

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  19 September 2016

Clifford Hurvich  and
Josh Reed
Clifford Hurvich*
Affiliation:
Leonard N. Stern School of Business New York University
Josh Reed*
Affiliation:
Leonard N. Stern School of Business New York University
*
* Postal address: Leonard N. Stern School of Business, New York University, 44 West 4th St., New York, NY 10012, USA.
* Postal address: Leonard N. Stern School of Business, New York University, 44 West 4th St., New York, NY 10012, USA.
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Abstract

We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

Keywords

Random walkqueueingheavy tail

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 744 - 767
Copyright
Copyright © Applied Probability Trust 2016 

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