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Maxima of independent, non-identically distributed Gaussian vectors. (English) Zbl 1322.60073

The authors consider independent triangular arrays \(\mathbf{X}_{i,n} = (X^{(1)}_{i,n}, \dots, X^{(d)}_{i,n})\), \(n \geq 1\), \(1 \leq i \leq n\), of zero-mean, unit-variance Gaussian random vectors with correlation matrix \(\Sigma_{i,n}\). Letting \(\mathbf{M}_n = (M^{(1)}_{n}, \dots, M^{(d)}_{n})\) denote the vector of component-wise maxima \(M^{(j)}_{n} = \max_{1\leq i \leq n}X^{(j)}_{i,n}, j \in \{1, \dots, d\}\), the convergence of the rescaled, row-wise maxima \(b_n(\mathbf{M}_n - b_n)\) is studied. Starting with bivariate triangular arrays, the authors introduce a sequence of counting measures to capture the dependence structure in each row and this is used to state criteria for the convergence. These results are used to completely characterize the max-limits of independent, but not necessarily identically distributed sequences of bivariate Gaussian vectors. The bivariate max-stable limit distributions are max-mixtures of Huesler-Reiss distributions with different dependency parameters and these constitute a large class of new bivariate max-stable distributions. Two of them, Rayleigh distributions and type-2 Gumbel distributions, are explicitly derived as examples. The multivariate case is, then derived and it is shown that the limit distributions arise as finite-dimensional marginals of the new class of max-mixtures of Brown-Resnick processes which offer a large variety of extremal correlation functions.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60G15 Gaussian processes
60F99 Limit theorems in probability theory
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