Extremes of multidimensional stationary Gaussian random fields

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 $P(\sup\{X(\mathbf{t}):\mathbf{t}\in[0,1]^d\}>u)^{-1}(1+o(1))$.