Extremes of Gaussian Random Fields with regularly varying dependence structure

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 $\mathbb{R}$, 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 $\mathcal{T}\subset {\mathbb{R}}^2$, 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.

38 pages