Extremes of homogeneous Gaussian random fields

Let $\{X(s,t):s,t\geqslant 0\}$ 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 $α_1,α_2\in(0,2],$ and $r(s,t)<1$ for $(s,t)\neq(0,0)$. In this contribution we derive an exact asymptotic expansion (as $u\to \infty$) of $\mathbb{P}\left(\sup_{(s n_1(u),t n_2(u))\in\left[0,x\right]\times\left[0,y\right]}X(s,t)\leqslant u\right),$ where $n_1(u)n_2(u)=u^{2/α_1+2/α_2}Ψ(u)$, which holds uniformly for $(x,y) \in [ A , B ]^2$ with $A , B$ two positive constants and $Ψ$ 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 $X(s,t)$.

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