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

Abstract

Let $\{X(t):t\in [0,\infty )\}$ be a centered stationary Gaussian process. We study the exact asymptotics of $\pr (\sup_{s \in [0,T]} X(t) > u)$, as $u\to \infty$, 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 $\lambda \in (0,1)$ and the case of $T$ having slowly varying tail distribution.