Extremes of Gaussian random fields with non-additive dependence structure

TL;DR

This paper derives exact asymptotics for the tail probabilities of the supremum of non-locally stationary Gaussian fields with complex dependence, extending understanding of extremes in high-dimensional stochastic processes.

Contribution

It provides the first precise asymptotic results for Gaussian fields with non-additive, non-locally stationary dependence structures, especially on Jordan sets with positive Lebesgue measure.

Findings

Exact asymptotics for tail probabilities of Gaussian field maxima.

Application to tail probabilities in performance tables and chi processes.

Extension to non-locally stationary dependence structures.

Abstract

We derive exact asymptotics of $$\mathbb{P}\left(\sup_{\mathbf{t}\in {\mathcal{A}}}X(\mathbf{t})>u\right),~ \text{as}~ u\to\infty,$$ for a centered Gaussian field $X(\mathbf{t}),~ \mathbf{t}\in \mathcal{A}\subset\mathbb{R}^n$, $n>1$ with continuous sample paths a.s., for which $\arg \max_{\mathbf{t}\in {\mathcal{A}}} Var(X(\mathbf{t}))$ is a Jordan set with finite and positive Lebesque measure of dimension $k\leq n$ and its dependence structure is not necessarily locally stationary. Our findings are applied to deriving the asymptotics of tail probabilities related to performance tables and chi processes where the covariance structure is not locally stationary.