On the ergodicity and mixing of max-stable processes

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Ergodic properties of max-infinitely divisible processes (2009, methodology): “Note also that the sequence r(t) is positive–definite (see [24] or [26]) and hence, by Bochner’s theorem, there is a symmetric with respect to 0 measure μ on [−π, π] (called the spectral measure of X) such that r(t) = ∫ π −π e dμ(x) for all t ∈ Z.”
Spectral representations of sum- and max-stable processes (2009, background): “By Theorem 3.2 of Stoev (2007), in order to show that η∨ is ergodic it suffices to show that for every f ∈ L1+(Ω,A, µ) lim T→∞ 1 T ∫ T 0 ‖f ∧ Utf‖1dt = 0 (12) (∧ denotes the minimum).”
Generalized Pickands constants and stationary max-stable processes (2016, background): “If {W (t), t ≤ 0} is defined as an exponentially transformed version of the corresponding {W (t), t ≥ 0}, then ξW can be shown to be stationary and max-stable; see [27, 28] for details.”
Statistical inference for max‐stable processes in space and time (2012, background): “Based on the extremal integral representation of max-stable processes Stoev (2008) establishes conditions under which a max-stable process is ergodic. Wang et al. (2011) extend these results to a”