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Extremes of Homogeneous Gaussian Random Fields

  • Krzysztof Dębicki (a1), Enkelejd Hashorva (a2) and Natalia Soja-Kukieła (a3)
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(sn 1(u),tn 2(u)) ∈[0,x]∙[0,y] X(s, t) ≤ u), where n 1(u)n 2(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).

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Corresponding author
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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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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