On Extremal Index of max-stable stationary processes

Abstract

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary process $W(t),t\in R$ as $$\mathcal{H}_W^\delta= \lim_{T\to\infty} T^{-1} E{ \left(\sup_{t\in \delta Z \cap [0,T]} e^{W(t)}\right) },\ \delta \ge 0$$ (set $0 Z=R$ if $\delta=0$) and the extremal index of the associated max-stable stationary process $\xi_W$. We derive several new formulas and obtain lower bounds for $\mathcal{H}_W^\delta$ if $W$ is a Gaussian or a L\'evy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.