Extremes of Gaussian random fields with non-additive dependence structure.

We derive the exact asymptotics of P{supt∈A X(t)>u} as u→∞, for a centered Gaussian field X(t), t ∈A⊂Rn, n>1 with continuous sample paths almost surely, for which arg maxt∈A Var(X(t)) is a Jordan set with a finite and positive Lebesgue measure of dimension k ≤ n and its dependence structure is not necessarily locally stationary. Our findings are applied to derive the asymptotics of tail probabilities related to performance tables and chi processes, particularly when the covariance structure is not locally stationary.