On the Structure and Representations of Max--Stable Processes

TL;DR

This paper classifies max-stable processes using spectral representations, exploring their structure and decompositions, and applies these results to examples like Brown-Resnick processes and Gaussian processes.

Contribution

It introduces a general classification framework for max-stable processes based on spectral functions and decompositions, connecting to flow theory and extending to Gaussian processes.

Findings

Brown-Resnick processes are dissipative.

Decompositions relate to non-singular flows.

Gaussian processes with stationary increments are analyzed.

Abstract

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.