Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let {X(t) : t is an element of vertical bar 0. infinity)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sup(s is an element of vertical bar 0,T vertical bar) X (s) > u), as u -> infinity, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T. the case of T having regularly varying tail distribution with parameter lambda is an element of (0, 1) and the case of T having slowly varying tail distribution. (C) 2011 Elsevier B.V. All rights reserved.