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Let $\{X(t) : t \in [0, \infty) \}$ be a centered stationary Gaussianprocess. 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 randomvariable. It appears that the heaviness of $T$ impacts the form of theasymptotics, leading to three scenarios: the case of integrable $T$, the caseof $T$ having regularly varying tail distribution with parameter$\lambda\in(0,1)$ and the case of $T$ having slowly varying tail distribution.

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