Ergodic decompositions of stationary max-stable processes in terms of their spectral functions
Résumé
We revisit conservative/dissipative and positive/null decomposi-tions 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 Cesàro 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.
Mots clés
Max-stable random process
de Haan representation
Non-singular flow
Conservative/dissipative decomposition
Positive/null decomposition
Ergodic process
Mixing process
Mixed moving maximum process
Max-stable random process
Mixed moving maximum process
Max-stable random process
Max-stable random process