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

Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fr'{e}chet marginals ${\mathrm{\Phi }}_{\alpha },\alpha >0$ and set $M_X(T)={\sup }_{t\in [0,T]}X(t),T>0$. In the light of the seminal articles [1,2], it follows that $A_T=M_X(T)\mathrm{/}{T}^{1\mathrm{/}\alpha }$ converges in distribution as $T\to \mathrm{\infty }$ to ${\mathcal{H}}_Z^{1\mathrm{/}\alpha }X(1)$, where ${\mathcal{H}}_Z$ is the Pickands constant corresponding to the spectral process $Z$ of $X$. In this contribution we derive explicit formulas for ${\mathcal{H}}_Z$ in terms of $Z$ and show necessary and sufficient conditions for its positivity. From our analysis it follows that $A_T^{\beta },T>0$ is uniformly integrable for any $\beta \in (0,\alpha)$. Further, we discuss the dissipative Rosi'nski (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.