Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

TL;DR

This paper derives the precise asymptotic behavior of the probability that a Gaussian process with stationary increments exceeds a high threshold over a random Weibullian time interval, with applications to fractional Laplace motion.

Contribution

It provides the first detailed asymptotic analysis of Gaussian supremum distributions over Weibullian random times, extending classical results to a new class of random horizons.

Findings

Derived exact asymptotics for Gaussian supremum over Weibullian times.

Applied results to fractional Laplace motion.

Enhanced understanding of Gaussian process extremes over random intervals.

Abstract

Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $\sigma^2_X(t)$. We study the exact asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.