Sojourns of Stationary Gaussian Processes over a Random Interval

We investigate asymptotics of the tail distribution of sojourn time integral(T)(0) I(X(t) > u)dt, as u -> infinity, where X is a centered stationary Gaussian process and T is an independent of X nonnegative random variable. The heaviness of the tail distribution of T impacts the form of the asymptotics, leading to four scenarios: the case of integrable T, the case of regularly varying T with index lambda = 1 and index lambda is an element of (0, 1) and the case of slowly varying tail distribution of T. The derived findings are illustrated by the analysis of the class of fractional Ornstein-Uhlenbeck processes.