Extremes of locally-homogenous vector-valued Gaussian processes

In this paper, we study the asymptotical behaviour of high exceedence probabilities for centered continuous \(\mathbb {R}^n\)-valued Gaussian random field \(\varvec{X}\) with covariance matrix satisfying \(\Sigma - R ( t + s, t ) \sim \sum _{l = 1}^n B_l ( t ) \, | s_l |^{\alpha _l}\) as \(s \downarrow 0\). Such processes occur naturally as time transformations of homogenous random fields, and we present two asymptotic results of this nature as applications of our findings. The technical novelty of our proof consists in showing that the Slepian-Gordon inequality technique, essential in the univariate case, can also be successfully applied in the multivariate setup. This is noteworthy because this technique was previously believed to be inaccessible in this particular context.