Sojourn times of Gaussian processes with trend

Abstract

We derive exact tail asymptotics of sojourn time above the level $u\ge 0$ $$\mathbb{P}\left(v(u){\int }_0^T\mathbb{I}(X(t)-ct>u)dt>x\right),x\ge 0$$ 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 $${\tau }_u(x):=\inf \left\{t\ge 0:v(u){\int }_0^t\mathbb{I}(X(s)-cs>u)ds>x\right\},$$ as $u\to \infty$, $x\ge 0$, where $\inf \varnothing =\infty$. 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.

30 pages