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

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