Extremes of multidimensional stationary Gaussian random fields

TL;DR

This paper analyzes the asymptotic behavior of the maximum of a multidimensional stationary Gaussian field over certain sets as the threshold goes to infinity, extending extreme value theory to high-dimensional Gaussian processes.

Contribution

It provides a detailed description of the asymptotics of the tail probability of the supremum of multidimensional Gaussian fields over specific sets, under general correlation decay conditions.

Findings

Derived asymptotic formulas for tail probabilities of maxima

Extended extreme value results to multidimensional Gaussian fields

Characterized the influence of correlation decay on extremes

Abstract

Let $\{X(\mathbf{t}):\mathbf{t}=(t_1, t_2, \ldots, t_d)\in[0,\infty)^d\}$ be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function $r$ satisfying conditions $r(\mathbf{t})<1$ for every $\mathbf{t}\neq \mathbf{0}$ and $r(\mathbf{t})=1-\sum_{i=1}^d |t_i|^{\alpha_i} + o(\sum_{i=1}^d |t_i|^{\alpha_i})$, as $\mathbf{t}\to\mathbf{0}$, with constants $\alpha_1, \alpha_2, \ldots, \alpha_d \in(0,2]$. The main result of this contribution is the description of the asymptotic behaviour of $P(\sup\{X(\mathbf{t}): \mathbf{t}\in\mathcal{J}^{\mathbf{x}}_{\mathbf{m}} \}\leqslant u)$, as $u\to\infty$, for some Jordan-measurable sets $\mathcal{J}^{\mathbf{x}}_{\mathbf{m}}$ of volume proportional to $P(\sup\{X(\mathbf{t}):\mathbf{t}\in[0,1]^d\}>u)^{-1}(1+o(1))$.