Extremes of vector-valued Gaussian processes: Exact asymptotics

Let {Xi (t), t ≥ 0}, 1 ≤ i ≤ n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of

P (∃t∈[0,T ]∀i=1,...,n Xi(t) > )

as u → ∞, for both locally stationary Xi’s and Xi’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.