Extremes of vector-valued Gaussian processes: exact asymptotics

Abstract

Let $\{X_i(t),t\ge 0\},1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$P\left({\exists }_{t\in [0,T]}{\forall }_{i=\mathrm{1...}n}{X}_i(t)>u\right)$$ as $u\to \infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

20 pages

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