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March 2016 Extremes of a class of nonhomogeneous Gaussian random fields
Krzysztof Dȩbicki, Enkelejd Hashorva, Lanpeng Ji
Ann. Probab. 44(2): 984-1012 (March 2016). DOI: 10.1214/14-AOP994


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.


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Krzysztof Dȩbicki. Enkelejd Hashorva. Lanpeng Ji. "Extremes of a class of nonhomogeneous Gaussian random fields." Ann. Probab. 44 (2) 984 - 1012, March 2016.


Received: 1 August 2013; Revised: 1 October 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

MathSciNet: MR3474465
Digital Object Identifier: 10.1214/14-AOP994

Primary: 60G15
Secondary: 60G70

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

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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