Maxima of independent, non-identically distributed Gaussian vectors

Let $X_{i,n}, n\in\mathbb{N}, 1\leq i \leq n$, be a triangular array of independent $\R^d$-valued Gaussian random vectors with covariance 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\"usler-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 limits of suitably normalized and randomly rescaled maxima of $n$ independent Gaussian processes whose finite dimensional margins coincide with the above limit distributions. 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.