Extremes for stationary regularly varying random fields over arbitrary index sets

TL;DR

This paper investigates the clustering behavior of extreme values in stationary regularly varying random fields over arbitrary index sets, extending existing theories to more general and complex index set geometries.

Contribution

It introduces new conditions for the existence of extremal limit fields over arbitrary index sets and extends spectral tail measures to describe clustering in general settings.

Findings

Limit of exceedance point fields exists under certain conditions.

Extremal dependence is local under anti-clustering conditions.

The $ u$-spectral tail measure generalizes clustering description.

Abstract

We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point random fields of the exceedances above a high threshold exists. Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature [3, 21]. However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure [23] except for hyperrectangles or index sets in the lattice case. Using the recent extension of the spectral measure for star-shaped equipped space [18], the $\upsilon$-spectral tail measure provides a natural extension that describes the clustering effect in full generality.