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

Abstract

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.

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