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Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be anon-negative constant. In this paper we study the asymptotics of$P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$,with $\mathcal{T}$ an independent of $X$ non-negative random variable. As anapplication, we derive the asymptotics of finite-time ruin probability oftime-changed fractional Brownian motion risk processes.

Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random Intervals