The Annals of Probability

Extremes of a class of nonhomogeneous Gaussian random fields

Krzysztof Dȩbicki, Enkelejd Hashorva, and Lanpeng Ji

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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

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

Received: August 2013
Revised: October 2014
First available in Project Euclid: 14 March 2016

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Mathematical Reviews number (MathSciNet)

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

Extremes nonhomogeneous Gaussian random fields Shepp statistics fractional Brownian motion maximum loss span of Gaussian processes Pickands constant Piterbarg constant generalized Pickands–Piterbarg constant


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.

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