On Extremal Index of Max-Stable Random Fields

TL;DR

This paper investigates the extremal index of stationary max-stable random fields, providing conditions for its possible values and showing its equivalence to the block extremal index, with implications for multivariate fields.

Contribution

It establishes necessary and sufficient conditions for the extremal index to be 0, positive, or 1, and demonstrates its equivalence to certain functional indices in max-stable fields.

Findings

Conditions for extremal index values are characterized.

Extremal index equals the block extremal index.

Functional indices coincide with the extremal index for many functionals.

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.