Approximation of Sojourn Times of Gaussian Processes

TL;DR

This paper studies the tail behavior of the time a Gaussian process spends above a threshold, providing new results for discrete-time cases and extending continuous-time analyses, along with a novel representation of Pickands constant.

Contribution

It introduces new asymptotic results for the sojourn times of Gaussian processes, especially in discrete-time, and offers a new representation of Pickands constant for simulations.

Findings

New tail asymptotics for discrete-time Gaussian processes

Extended continuous-time results for non-stationary processes

A sharp lower bound for Pickands constant

Abstract

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes $X$, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of [1,2] for non-stationary $X$. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.