Maxima of independent, non-identically distributed Gaussian vectors

Abstract

Let $X_{i,n},n\in \mathbb{N},1\le i\le n$, 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.

Published at http://dx.doi.org/10.3150/13-BEJ560 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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