Extremes of multidimensional stationary Gaussian random fields

Abstract

Let $\{X(\mathbf{t}):\mathbf{t}=(t_1,t_2,\dots ,t_d)\in [0,\infty {)}^d\}$ 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 $\mathbf{t}\ne 0$ and $r(\mathbf{t})=1-{\sum }_{i=1}^d|t_i{|}^{{\alpha }_i}+o({\sum }_{i=1}^d|t_i{|}^{{\alpha }_i})$, as $\mathbf{t}\to 0$, with constants ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_d\in (0,2]$. The main result of this contribution is the description of the asymptotic behaviour of $P(\sup \{X(\mathbf{t}):\mathbf{t}\in {\mathcal{J}}_{\mathbf{m}}^{\mathbf{x}}\}⩽u)$, as $u\to \infty$, for some Jordan-measurable sets ${\mathcal{J}}_{\mathbf{m}}^{\mathbf{x}}$ of volume proportional to $P(\sup \{X(\mathbf{t}):\mathbf{t}\in [0,1{]}^d\}>u{)}^{-1}(1+o(1))$.