Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants

Let $$X(t),t\in \mathbb {R}$$X(t),t∈R be a stochastically continuous stationary max-stable process with Fréchet marginals $$\Phi _\alpha , \alpha >0$$Φα,α>0 and set $$M_X(T)=\sup _{t \in [0,T]} X(t),T>0$$MX(T)=supt∈[0,T]X(t),T>0. In the light of the seminal articles (Samorodnitsky in Ann Probab 32(2):1438–1468, 2004; Adv Appl Probab 36(3):805–823, 2004), it follows that $$A_T=M_X(T)/T^{1/\alpha }$$AT=MX(T)/T1/α converges in distribution as $$T\rightarrow \infty $$T→∞ to $$ \mathcal {H}^{1/\alpha } X(1)$$H1/αX(1), where $$ \mathcal {H}$$H is the Pickands constant corresponding to the spectral process Z of X. In this contribution, we derive explicit formulas for $$ \mathcal {H}$$H in terms of Z and show necessary and sufficient conditions for its positivity. From our analysis, it follows that $$A_T^\beta ,T>0$$ATβ,T>0 is uniformly integrable for any $$\beta \in (0,\alpha )$$β∈(0,α). For Brown–Resnick X, we show the validity of the celebrated Slepian inequality and discuss the finiteness of Piterbarg constants.