Extremes of Gaussian Random Fields with regularly varying dependence structure

2016/05/28 by Dębicki, Krzyztof, Hashorva, Enkelejd, Liu, Peng
#FOS: Mathematics #Primary 60G15 #Probability (math.PR) #secondary 60G70

paper · doi:10.48550/arxiv.1605.08946

Abstract

Let X(t), t∈ T be a centered Gaussian random field with variance function σ²(⋅) that attains its maximum at the unique point t₀∈ T, and let M(T):=sup_{t∈ T} X(t). For T a compact subset of \R, the current literature explains the asymptotic tail behaviour of M(T) under some regularity conditions including that 1- σ(t) has a polynomial decrease to 0 as t → t₀. In this contribution we consider more general case that 1- σ(t) is regularly varying at t₀. We extend our analysis to random fields defined on some compact T⊂ \R², deriving the exact tail asymptotics of M(T) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

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