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 $${\mathcal{H}}_W^{\delta }=\underset{T\to \infty }{\lim }{T}^{-1}E\left(\underset{t\in \delta Z\cap [0,T]}{\sup }{e}^{W(t)}\right),\delta \ge 0$$ (set $0Z=R$ 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 Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

24 pages

References in corpus (1)

• Extremes of $q$-Ornstein-Uhlenbeck processes

Cited by in corpus (4)

• Uniform tail approximation of homogenous functionals of Gaussian fields
• On the speed of convergence of discrete Pickands constants to continuous ones
• Approximation of Sojourn Times of Gaussian Processes
• Shift-invariant homogeneous classes of random fields