Approximation of Supremum of Max-Stable Stationary Processes and Pickands Constants
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Let X(t),t∈R be a stochastically continuous stationary max-stable process with Fréchet marginals Φ_α, α>0 and set M_X(T)=_t ∈ [0,T] X(t),T>0. In the light of the seminal articles [1,2], it follows that A_T=M_X(T)/T^1/α converges in distribution as T→∞ to H_Z^1/α X(1), where H_Z is the Pickands constant corresponding to the spectral process Z of X. In this contribution we derive explicit formulas for H_Z in terms of Z and show necessary and sufficient conditions for its positivity. From our analysis it follows that A_T^β,T>0 is uniformly integrable for any β∈ (0,α). Further, we discuss the dissipative Rosiński (or mixed moving maxima) representation of X. Additionally, for Brown-Resnick X we show the validity of the celebrated Slepian inequality and obtain lower bounds on the growth of supremum of Gaussian processes with stationary increments by exploiting the link between Pickands constants and Wills functional. Moreover, we derive upper bounds for supremum of centered Gaussian processes given in terms of Wills functional, and discuss the relation between Pickands and Piterbarg constants.
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