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November 2017 Convergence of the centered maximum of log-correlated Gaussian fields
Jian Ding, Rishideep Roy, Ofer Zeitouni
Ann. Probab. 45(6A): 3886-3928 (November 2017). DOI: 10.1214/16-AOP1152


We show that the centered maximum of a sequence of logarithmically correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of branching random walk and Gaussian chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields; some additional structural assumptions of the type we make are needed for convergence of the centered maximum.


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Jian Ding. Rishideep Roy. Ofer Zeitouni. "Convergence of the centered maximum of log-correlated Gaussian fields." Ann. Probab. 45 (6A) 3886 - 3928, November 2017.


Received: 1 April 2015; Revised: 1 July 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838110
MathSciNet: MR3729618
Digital Object Identifier: 10.1214/16-AOP1152

Primary: 60G15 , 60G60 , 60G70

Keywords: extremes values , Gaussian processes , log-correlated fields

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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