## The Annals of Probability

### Extremes of a class of nonhomogeneous Gaussian random fields

#### 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.

#### Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 984-1012.

Dates
Revised: October 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960389

Digital Object Identifier
doi:10.1214/14-AOP994

Mathematical Reviews number (MathSciNet)
MR3474465

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes

#### Citation

Dȩbicki, Krzysztof; Hashorva, Enkelejd; Ji, Lanpeng. Extremes of a class of nonhomogeneous Gaussian random fields. Ann. Probab. 44 (2016), no. 2, 984--1012. doi:10.1214/14-AOP994. https://projecteuclid.org/euclid.aop/1457960389

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