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

Let {X(t):t∈0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sup_s∈0,T]X(s)>u), as u→∞, 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 λ∈(0,1) and the case of T having slowly varying tail distribution.