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Equivalent representations of max-stable processes via ℓp-norms

Part of: Stochastic processes

Published online by Cambridge University Press:  28 March 2018

Marco Oesting
Marco Oesting*
Affiliation:
Universität Siegen
*
* Postal address: Department Mathematik, Universität Siegen, Walter-Flex-Str. 3, 57072 Siegen, Germany. Email address: oesting@mathematik.uni-siegen.de
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Abstract

While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on ℓp-norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of an ℓp-norm-based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.

Keywords

Extreme value theoryReich–Shaby modelspectral representationstable tail dependence function

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G52: Stable processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 54 - 68
Copyright
Copyright © Applied Probability Trust 2018 

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