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Sparre Andersen identity and the last passage time

Part of: Stochastic processes

Published online by Cambridge University Press:  21 June 2016

Jevgenijs Ivanovs
Jevgenijs Ivanovs*
Affiliation:
University of Lausanne
*
* Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, Switzerland. Email address: jevgenijs.ivanovs@unil.ch
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Abstract

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It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in (-∞, 0], say σ, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on [0, σ]. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

Keywords

Last passagetime reversalexchangeable incrementsuniform law

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G50: Sums of independent random variables; random walks
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 600 - 605
Copyright
Copyright © Applied Probability Trust 2016 

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