Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals

Let {X(t),t ≥ 0} be a centered Gaussian process and let γ be a non-negative constant. In this paper we study the asymptotics of \(\mathbb {P} \left \{\underset {t\in 0,\mathcal {T}/u^{\gamma }]}\sup X(t)>u\right \}\) as \(u\rightarrow \infty \) , 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.