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Extremes of Homogeneous Gaussian Random Fields

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Krzysztof Dębicki ,
Enkelejd Hashorva  and
Natalia Soja-Kukieła
Krzysztof Dębicki*
Affiliation:
University of Wrocław
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Natalia Soja-Kukieła*
Affiliation:
Nicolaus Copernicus University
*
∗ Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: debicki@math.uni.wroc.pl
∗∗ Postal address: Faculty of Business and Economics (HEC Lausanne), University of Lausanne, 1015 Lausanne, Switzerland.
∗∗∗ Postal address: Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland.
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Abstract

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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).

Keywords

Gaussian random fieldsupremumtail asymptoticyextremal indexBerman conditionstrong dependence

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 55 - 67
Copyright
© Applied Probability Trust 

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