Extremes for stationary regularly varying random fields over arbitrary index sets

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 υ-spectral tail measure provides a natural extension that describes the clustering effect in full generality.

Extremes of regularly varying stochastic volatility fields

“…and thus, to show (22), it suffices to show that lim sup (in n and t) of each sum equals zero. We only consider the first sum in (23), as the second sum is handled identically.…”

“…Results on extremes of stationary random fields are to the best of the authors' knowledge mostly formulated under the assumption of (D n ) being a sequence of increasing boxes ( [13,20,24,29]). However, there are recent papers in which extremes are considered under a more general geometric regime; see for instance [23,25,26]. In this paper we simply assume that D n ⊆ Z d can be approximated from the in-and outside by unions of certain slowly increasing boxes, where the approximations asymptotically have the same size as D n ; please see Assumption 1 for details.…”