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

Abstract: Let ${X(t),t\ge0}$ be a centered Gaussian process and let $\gamma$ be a non-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 an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.