Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statistics

TL;DR

This paper derives precise tail asymptotics and extreme value laws for a class of locally stationary Gaussian fields, with applications to Shepp statistics involving various Gaussian processes.

Contribution

It provides the first exact tail asymptotics and extreme limit laws for maxima of a specific class of Gaussian fields with applications to Shepp statistics.

Findings

Exact tail asymptotics for the maximum of Gaussian fields.

Extreme limit laws for the maxima of these fields.

Applications to Shepp statistics with different Gaussian processes.

Abstract

Let $\{Z(\tau,s), (\tau,s)\in [a,b]\times[0,T]\}$ with some positive constants $a,b,T$ be a centered Gaussian random field with variance function $\sigma^{2}(\tau,s)$ satisfying $\sigma^{2}(\tau,s)=\sigma^{2}(\tau)$. We firstly derive the exact tail asymptotics for the maximum $M_{H}(T)=\max_{(\tau,s)\in[a,b]\times[0,T]}Z(\tau,s)/\sigma(\tau)$ up crossing some level $u$ with any fixed $0<a<b<\infty$ and $T>0$; and we further derive the extreme limit law for $M_{H}(T)$. As applications of the main results, we derive the exact tail asymptotics and the extreme limit law for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as input.