Extremes of homogeneous Gaussian random fields. (English) Zbl 1315.60059

Dȩbicki, Krzysztof; Hashorva, Enkelejd; Soja-Kukieła, Natalia

J. Appl. Probab. 52, No. 1, 55-67 (2015).

Summary: Let \(\{X(s,t): s, t\geq 0\}\) be a centered homogeneous Gaussian field with almost surely continuous sample paths and correlation function \(r(s,t)=\text{cov}(X(s,t), X(0,0))\) such that \(r(s,t)=1-|s|^{\alpha_1}-|t|^{\alpha_2}+o(|s|^{\alpha_1}+|t|^{\alpha_2})\), \(s,t\to 0\), with \(\alpha_1, \alpha_2 \in (0, 2]\), and \(r(s,t)<1\) for \((s,t)\neq (0,0)\). In this contribution we derive an asymptotic expansion (as \(u\to \infty\)) of \(\mathbb{P}(\sup_{(sn_1(u),tn_2(u))\in [0,x]\times [0,y]}X(s,t)\leq u)\), 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 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 \(X(s,t)\).

MSC:

60G60	Random fields
60G70	Extreme value theory; extremal stochastic processes
60G15	Gaussian processes

Keywords:

Gaussian random fields; tail asymptotics; extremal index; Berman condition; strong dependence

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