March 2015 Extremes of homogeneous Gaussian random fields
Krzysztof Dębicki, Enkelejd Hashorva, Natalia Soja-Kukieła
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J. Appl. Probab. 52(1): 55-67 (March 2015). DOI: 10.1239/jap/1429282606


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) = u2/α1+2/α2Ψ(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).


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Krzysztof Dębicki. Enkelejd Hashorva. Natalia Soja-Kukieła. "Extremes of homogeneous Gaussian random fields." J. Appl. Probab. 52 (1) 55 - 67, March 2015.


Published: March 2015
First available in Project Euclid: 17 April 2015

zbMATH: 1315.60059
MathSciNet: MR3336846
Digital Object Identifier: 10.1239/jap/1429282606

Primary: 60G15
Secondary: 60G70

Keywords: Berman condition , extremal index , Gaussian random field , Strong dependence , supremum , tail asymptoticy

Rights: Copyright © 2015 Applied Probability Trust


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Vol.52 • No. 1 • March 2015
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