Sojourn times of Gaussian processes with trend

TL;DR

This paper derives exact asymptotics for the tail probabilities of sojourn times of Gaussian processes with trend, and analyzes the asymptotic distribution of the corresponding first passage times as the level tends to infinity.

Contribution

It provides new exact tail asymptotics and distributional properties for sojourn times of Gaussian processes with trend, extending previous results to a broader class of processes.

Findings

Exact tail asymptotics for Gaussian processes with trend

Asymptotic distribution of first passage times

Application to stationary increment and self-similar processes

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 $$\tau_u(x):=\inf\left\{t\geq 0: v(u) \int_0^t \mathbb{I}(X(s)-cs>u)d s>x\right\}, $$ 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.