Maxima of independent, non-identically distributed Gaussian vectors

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 Hüsler–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.