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

Abstract

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.