Extremal t processes: Elliptical domain of attraction and a spectral representation

TL;DR

This paper introduces a spectral representation for the extremal t process, enabling direct simulation and highlighting its role as a maximum attractor for elliptical distributions, thus advancing spatial extremes modeling.

Contribution

It provides the first spectral construction for the extremal t process, including the extremal Gaussian as a special case, and clarifies its domain of attraction.

Findings

Spectral representation enables direct simulation of extremal t processes.

Extremal Gaussian process is a special case of extremal t process.

Extremal t process is the maximum domain of attraction for elliptical distributions.

Abstract

The extremal t process was proposed in the literature for modeling spatial extremes within a copula framework based on the extreme value limit of elliptical t distributions (Davison, Padoan and Ribatet (2012)). A major drawback of this max-stable model was the lack of a spectral representation such that for instance direct simulation was infeasible. The main contribution of this note is to propose such a spectral construction for the extremal t process. Interestingly, the extremal Gaussian process introduced by Schlather (2002) appears as a special case. We further highlight the role of the extremal t process as the maximum attractor for processes with finite-dimensional elliptical distributions. All results naturally also hold within the multivariate domain.