Exact asymptotics of supremum of a stationary Gaussian process over a random interval

TL;DR

This paper derives the exact asymptotic probabilities for the supremum of a stationary Gaussian process over a random interval, revealing how the tail behavior of the interval length influences the asymptotics.

Contribution

It provides a comprehensive analysis of the asymptotics of the supremum probability over a random interval, considering different tail behaviors of the interval length distribution.

Findings

Asymptotics differ based on the tail heaviness of the random interval T.

Three distinct scenarios identified: integrable T, regularly varying T, and slowly varying T.

Explicit asymptotic formulas derived for each case.

Abstract

Let $\{X(t) : t \in [0, \infty) \}$ be a centered stationary Gaussian process. We study the exact asymptotics of $\pr (\sup_{s \in [0,T]} X(t) > u)$, as $u \to \infty$, where $T$ is an independent of \{X(t)\} nonnegative random variable. It appears that the heaviness of $T$ impacts the form of the asymptotics, leading to three scenarios: the case of integrable $T$, the case of $T$ having regularly varying tail distribution with parameter $\lambda\in(0,1)$ and the case of $T$ having slowly varying tail distribution.