Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

TL;DR

This paper provides simple, spectral-function-based criteria to decompose stationary max-stable processes into conservative/dissipative and positive/null parts, and characterizes properties like ergodicity and mixing.

Contribution

It introduces new criteria for spectral functions to determine ergodic and mixing properties, simplifying the analysis of max-stable processes.

Findings

Spectral function is null-recurrent iff it converges to 0 in Cesàro sense.

Spectral function is dissipative iff it converges to 0 for processes with bounded paths.

Spectral function is integrable iff it converges to 0 almost surely.

Abstract

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property.