CONVERGENCE OF THE CENTERED MAXIMUM OF LOG-CORRELATED GAUSSIAN FIELDS

• 成果类型: Article
• 署名作者: Ding, Jian; Roy, Rishideep; Zeitouni, Ofer
• 署名单位: University of Chicago; Indian Institute of Management (IIM System); Indian Institute of Management Bangalore; Weizmann Institute of Science; New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1152
• 发表日期: 2017
• 页码: 3886-3928
• 关键词: LAW

摘要

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.