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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:127

Article

Dombry, Clement¹  Kabluchko, Zakhar² 

[1] Univ Bourgogne Franche Comte, CNRS, Lab Math Besancon, UMR 6623, 16 Route Gray, F-25030 Besancon, France

[2] Univ Munster, Inst Math Stat, Orleans Ring 10, D-48149 Munster, Germany

关键词: Max-stable random process;    de Haan representation;    Non-singular flow;    Conservative/dissipative decomposition;    Positive/null decomposition;    Ergodic process;    Mixing process;    Mixed moving maximum process;   

DOI  :  10.1016/j.spa.2016.10.001

来源: Elsevier

【 摘 要 】

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 if it converges to 0 in the Cesaro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative if it converges to 0. Surprisingly, for such processes a spectral function is integrable a.s. if 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. (C) 2016 Elsevier B.V. All rights reserved.

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