Bernoulli

Maxima of independent, non-identically distributed Gaussian vectors

Sebastian Engelke, Zakhar Kabluchko, and Martin Schlather

Full-text: Open access

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.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 38-61.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597063

Digital Object Identifier
doi:10.3150/13-BEJ560

Mathematical Reviews number (MathSciNet)
MR3322312

Zentralblatt MATH identifier
1322.60073

Keywords
extremal correlation function Gaussian random vectors Hüsler–Reiss distributions max-limit theorems max-stable distributions triangular arrays

Citation

Engelke, Sebastian; Kabluchko, Zakhar; Schlather, Martin. Maxima of independent, non-identically distributed Gaussian vectors. Bernoulli 21 (2015), no. 1, 38--61. doi:10.3150/13-BEJ560. https://projecteuclid.org/euclid.bj/1426597063


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