Extremes of a class of nonhomogeneous Gaussian random fields

TL;DR

This paper derives exact tail asymptotics for the maximum of a broad class of nonhomogeneous Gaussian random fields, extending classical results and applying to various Gaussian processes and their extremal statistics.

Contribution

It establishes the exact tail asymptotics for nonhomogeneous Gaussian fields with variance maxima on segments, extending classical homogeneous field results.

Findings

Exact tail asymptotics for nonhomogeneous Gaussian fields

Applications to Shepp statistics, Brownian bridge, and fractional Brownian motion

Asymptotic expansion for maximum loss and span of stationary Gaussian processes

Abstract

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.