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The extremes of random walks in random sceneries

Published online by Cambridge University Press:  01 July 2016

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.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2009 

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