Extremes of Gaussian processes over an infinite horizon

Consider a centered separable Gaussian process $Y$ with a variance function that is regularly varying at infinity with index $2H \in (0,2).$ Let $\phi$ be a ‘drift’ function that is strictly increasing, regularly varying at infinity with index $\beta > H,$ and vanishing at the origin. Motivated by queueing and risk models, we investigate the asymptotics for $u \to \infty$ of the probability $P(\sup_{t\geq 0}Y_t - \phi(t)>u).$ To obtain the asymptotics, we tailor the celebrated double sum method to our general framework. Two different families of correlation structures are studied, leading to four qualitatively different types of asymptotic behavior. A generalized Pickands’ constant appears in one of these cases. Our results cover both processes with stationary increments (including Gaussian integrated processes) and self-similar processes.