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

Clément Dombry and Zakhar Kabluchko

Stochastic Processes and their Applications, 2017, vol. 127, issue 6, 1763-1784

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 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.

Keywords: Max-stable random process; de Haan representation; Non-singular flow; Conservative/dissipative decomposition; Positive/null decomposition; Ergodic process; Mixing process; Mixed moving maximum process (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (5)

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DOI: 10.1016/j.spa.2016.10.001

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