On Extremal Index of Max-Stable Random Fields

Abstract

For a given stationary max-stable random field $X(t),t\in Z^d$ the corresponding generalised Pickands constant coincides with the classical extremal index $\theta$ which always exists. In this contribution we discuss necessary and sufficient conditions for $\theta$ to be 0, positive or equal to 1 and also show that $\theta$ is equal to the so-called block extremal index. Further, we consider some general functional indices of $X$ and prove that for a large class of functionals they coincide with $\theta$. Our study of max-stable and stationary random fields is important since the formulas are valid with obvious modifications for the candidate extremal index of multivariate regularly varying random fields.

arXiv commentMinor revision, to appear in Lithuanian Math Journal

References in corpus (5)

• Stationary max-stable fields associated to negative definite functions
• Detecting simultaneous variant intervals in aligned sequences
• On asymptotic constants in the theory of extremes for Gaussian processes
• Invariance properties of random vectors and stochastic processes based on the zonoid concept
• The distribution of maxima of approximately Gaussian random fields

Cited by in corpus (3)

• Tail Measures and Regular Variation
• Shift-invariant homogeneous classes of random fields
• Multivariate Max-Stable Processes and Homogeneous Functionals