Extremes of Homogeneous Gaussian Random Fields

Abstract

Let {X(s, t) : s, t 0} be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function r(s, t) = Cov(X(s, t), X(0, 0)) such thatwith α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) = (0, 0). In this contribution we derive an exact asymptotic expansion (as u → ∞) ofwhere n1(u)n2(u) = u 2/α 1 +2/α 2 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 asymptotics 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).

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

“…Leadbetter et al (1983), Pickands (1969) and Piterbarg (1996). The limit distribution theorem about M T extended by Mittal and Ylvisaker (1975) and McCormick and Qi (2000), and Dȩbicki et al (2013) extended the results to homogeneous Gaussian random fields.…”

Extremes of standard multifractional Brownian motion

“…These constants defined above play a significant role in the following theorems, see [9] for various properties of these constants and compare with, e.g., [10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

On extremal index of max-stable stationary pro­cesses

“…For instance in sequential analysis and statistical applications [43,44] and extremes of random fields [52,31] just to mention a few. For large classes of Gaussian rf's extremal indices have been discussed in [26,11,45], see also [49,4] for non-Gaussian cases and related results.…”