Extremes of Gaussian random fields with regularly varying dependence structure

Let (X(t), tin mathcal {T}) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point (t_{0}in mathcal {T}), and let (M(mathcal {T})=sup _{tin mathcal {T}} X(t)). For (mathcal {T}) a compact subset of ?, the current literature explains the asymptotic tail behaviour of (M(mathcal {T})) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set (mathcal {T}subset mathbb {R}^{2}), deriving the exact tail asymptotics of (M(mathcal {T})) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. 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.