Maxima of independent, non-identically distributed Gaussian vectors

TL;DR

This paper investigates the conditions under which maxima of independent Gaussian vectors converge to max-stable distributions, introduces new models for bivariate extremes, and constructs a broad class of stationary max-stable processes with flexible dependence structures.

Contribution

It provides necessary and sufficient conditions for convergence to max-stable distributions, introduces new models for bivariate extremes, and defines a new class of stationary max-stable processes as max-mixtures.

Findings

Necessary conditions for maxima convergence to max-stable distributions.

Explicit new models for bivariate extremes.

A broad class of max-stable processes with flexible extremal correlation functions.

Abstract

Let $X_{i,n},n\in \mathbb{N},1\leq i\leq n$, be a triangular array of independent $\mathbb{R}^d$-valued Gaussian random vectors with correlation matrices $\Sigma_{i,n}$. We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of H\"{u}sler-Reiss distributions. In the bivariate case, the conditions will also be sufficient. Using these results, new models for bivariate extremes are derived explicitly. Moreover, we define a new class of stationary, max-stable processes as max-mixtures of Brown-Resnick processes. As an application, we show that these processes realize a large set of extremal correlation functions, a natural dependence measure for max-stable processes. This set includes all functions $\psi(\sqrt{\gamma(h)}),h\in \mathbb{R}^d$, where $\psi$ is a completely monotone function and $\gamma$ is an arbitrary variogram.