Sojourn times of Gaussian processes with trend

Abstract

We derive exact tail asymptotics of sojourn time above the level $u\geq 0$ $$ \mathbb{P}\left(v(u)\int_0^T \mathbb{I}(X(t)-ct>u)d t>x\right), \quad x\geq 0 $$ as $u\to\infty$, where $X$ is a Gaussian process with continuous sample paths, $c>0$, $v(u)$ is a positive function of $u$ and $T\in (0,\infty]$. Additionally, we analyze asymptotic distributional properties of $$τ_u(x):=\inf\left\{t\geq 0: v(u) \int_0^t \mathbb{I}(X(s)-cs>u)d s>x\right\}, $$ as $u\to\infty$, $x\geq 0$, where $\inf\emptyset=\infty$. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar processes.