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

Let {X(t):t∈[0,∞)}{X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sups∈[0,T]X(s)>u)P(sups∈[0,T]X(s)>u), as u→∞u→∞, where TT is an independent of {X(t)}{X(t)} nonnegative random variable. It appears that the heaviness of TT impacts the form of the asymptotics, leading to three scenarios: the case of integrable TT, the case of TT having regularly varying tail distribution with parameter λ∈(0,1)λ∈(0,1) and the case of TT having slowly varying tail distribution.