On the Structure and Representations of Max--Stable Processes

Details On the Structure and Representations of Max--Stable Processes On the Structure and Representations of Max--Stable Processes

2009-03-20

arxiv.org/abs/0903.3594v2

Mathematics

Probability

40 pages. Minor changes. Technical Report 487, Department of Statistics, University of Michigan

Scientific paper

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 positive--null decompositions. For stationary max--stable processes, the latter two decompositions arise from connections to non--singular 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. A result on general Gaussian processes with stationary increments and continuous paths is obtained.

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