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On the structure and representations of max-stable processes

Part of: Ergodic theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Yizao Wang  and
Stilian A. Stoev
Yizao Wang*
Affiliation:
University of Michigan
Stilian A. Stoev*
Affiliation:
University of Michigan
*
∗ Postal address: Department of Statistics, The University of Michigan, 439 W. Hall, 1085 S. University, Ann Arbor, MI 48109-1107, USA.
∗ Postal address: Department of Statistics, The University of Michigan, 439 W. Hall, 1085 S. University, Ann Arbor, MI 48109-1107, USA.
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Abstract

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We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max-stable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

Keywords

Max-stable processclassificationmax-linear isometryspectral representationnonsingular flowco-spectral function

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G10: Stationary processes 60G52: Stable processes 37A50: Relations with probability theory and stochastic processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 855 - 877
Copyright
Copyright © Applied Probability Trust 2010 

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