## The Annals of Probability

### Convergence of the centered maximum of log-correlated Gaussian fields

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3886-3928.

Dates
Revised: July 2016
First available in Project Euclid: 27 November 2017

https://projecteuclid.org/euclid.aop/1511773667

Digital Object Identifier
doi:10.1214/16-AOP1152

Mathematical Reviews number (MathSciNet)
MR3729618

Zentralblatt MATH identifier
06838110

#### Citation

Ding, Jian; Roy, Rishideep; Zeitouni, Ofer. Convergence of the centered maximum of log-correlated Gaussian fields. Ann. Probab. 45 (2017), no. 6A, 3886--3928. doi:10.1214/16-AOP1152. https://projecteuclid.org/euclid.aop/1511773667

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