Extremes of Gaussian Random Fields with regularly varying dependence structure

TL;DR

This paper investigates the extreme value behavior of Gaussian random fields with regularly varying dependence structures, extending existing results to more general variance decay and higher-dimensional settings, including non-locally additive dependencies.

Contribution

It extends asymptotic tail analysis of Gaussian fields to cases with regularly varying variance decay and applies to 2D fields, including non-locally additive dependence structures.

Findings

Derived exact tail asymptotics for Gaussian fields with regularly varying variance.

Extended analysis to 2D Gaussian fields with complex dependence structures.

Identified new asymptotic behaviors for non-locally additive Gaussian fields.

Abstract

Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $\sigma^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For $\mathcal{T}$ a compact subset of $\R$, the current literature explains the asymptotic tail behaviour of $M(\mathcal{T})$ under some regularity conditions including that $1- \sigma(t)$ has a polynomial decrease to 0 as $t \to t_0$. In this contribution we consider more general case that $1- \sigma(t)$ is regularly varying at $t_0$. We extend our analysis to random fields defined on some compact $\mathcal{T}\subset \R^2$, deriving the exact tail asymptotics of $M(\mathcal{T})$ for the class of Gaussian random fields with variance and correlation functions being regularly varying at $t_0$. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.