Convergence of the centered maximum of log-correlated Gaussian fields

• Jian Ding ^[1] ; Rishideep Roy ^[3] ; Ofer Zeitouni ^[2]
  1. [1] University of Chicago

     University of Chicago

     City of Chicago, Estados Unidos

     
  2. [2] New York University

     New York University

     Estados Unidos

     
  3. [3] Indian Institute of Management Bangalore(India)

  Mostrar afiliaciones +
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 1, 2017, págs. 3886-3928
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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.