Extremes of homogeneous Gaussian random fields

Abstract

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|^α₁ - |t|^α₂ + o(|s|^α₁ + |t|^α₂), s, t → 0, with α₁, α₂ ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup_{(sn1(u),tn2(u)) ∈[0,x]∙[0,y]}X(s, t) ≤ u), where n₁(u)n₂(u) = u^{2/α₁+2/α₂}Ψ(u), which holds uniformly for (x, y) ∈ [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 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).