  Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian
processes with almost surely continuous sample paths. We derive the exact
asymptotics of $$ P\left(\exists_{t \in [0,T]}
  \forall_{i=1 ... n} X_i(t)> u \right) $$ 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.
