Extremes of homogeneous Gaussian random fields

Abstract

Let $\{X(s,t):s,t⩾0\}$ be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function $r(s,t)=Cov(X(s,t),X(0,0))$ such that \[r(s,t)=1-|s|^{α_1}-|t|^{α_2}+o(|s|^{α_1}+|t|^{α_2}), \quad s,t \to 0,\] with ${\alpha }_1,{\alpha }_2\in (0,2],$ and $r(s,t)<1$ for $(s,t)\ne (0,0)$. In this contribution we derive an exact asymptotic expansion (as $u\to \infty$) of $$\mathbb{P}\left(\underset{(s{n}_1(u),t{n}_2(u))\in \left[0,x\right]\times \left[0,y\right]}{\sup }X(s,t)⩽u\right),$$ where $n_1(u)n_2(u)=u^{2/{\alpha }_1+2/{\alpha }_2}\Psi (u)$, which holds uniformly for $(x,y)\in [A,B{]}^2$ with $A,B$ two positive constants and $\Psi$ the survival function of an $N(0,1)$ random variable. We apply our findings to the analysis of asymptotics of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally we determine the extremal index of the discretised random field determined by $X(s,t)$.

17 pages