September 2016 Series expansions for the all-time maximum of α-stable random walks
Clifford Hurvich, Josh Reed
Author Affiliations +
Adv. in Appl. Probab. 48(3): 744-767 (September 2016).

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.

Citation

Download Citation

Clifford Hurvich. Josh Reed. "Series expansions for the all-time maximum of α-stable random walks." Adv. in Appl. Probab. 48 (3) 744 - 767, September 2016.

Information

Published: September 2016
First available in Project Euclid: 19 September 2016

zbMATH: 1351.60054
MathSciNet: MR3568890

Subjects:
Primary: 60G50
Secondary: 60K25

Keywords: credit risk , Doubly stochastic process , multi-state life insurance models , stochastic interest , stochastic mortality

Rights: Copyright © 2016 Applied Probability Trust

JOURNAL ARTICLE
24 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.48 • No. 3 • September 2016
Back to Top