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The extremes of random walks in random sceneries

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Brice Franke  and
Tatsuhiko Saigo
Brice Franke*
Affiliation:
Ruhr-Universität Bochum
Tatsuhiko Saigo*
Affiliation:
National Taiwan University
*
∗ Postal address: Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150, 44780 Bochum, Germany. Email address: [email protected]
∗∗ Current address: Department of Mathematics, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa-ken prefecture, 223-8522, Japan. Email address: [email protected]
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Abstract

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In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

Keywords

Extremesstationary sequencesrandom walkrandom scenery

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 452 - 468
Copyright
Copyright © Applied Probability Trust 2009 

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