Extremes of Gaussian random fields with nonadditive dependence structure

Abstract We derive the exact asymptotics of $\mathbb{P} {\{\sup\nolimits_{\boldsymbol{t}\in {\mathcal{A}}}X(\boldsymbol{t})>u \}} \textrm{ as}\ u\to\infty,$ for a centered Gaussian field $X({\boldsymbol{t}}),\ {\boldsymbol{t}}\in \mathcal{A}\subset\mathbb{R}^n$ , $n>1$ with continuous sample paths almost surely, for which $\arg \max_{\boldsymbol{t}\in {\mathcal{A}}} {\mathrm{Var}}(X(\boldsymbol{t}))$ is a Jordan set with a finite and positive Lebesgue measure of dimension $k\le n$ 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.