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Mod-Gaussian convergence: new limit theorems in probability and number theory


Jacod, J; Kowalski, E; Nikeghbali, A (2011). Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Mathematicum, 23(4):835-873.

Abstract

We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős-Kac Theorem.

Abstract

We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős-Kac Theorem.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Physical Sciences > Applied Mathematics
Language:English
Date:2011
Deposited On:09 Jan 2012 15:07
Last Modified:06 Feb 2024 02:41
Publisher:De Gruyter
ISSN:0933-7741
OA Status:Green
Publisher DOI:https://doi.org/10.1515/form.2011.030
Related URLs:http://arxiv.org/abs/0807.4739
  • Content: Published Version
  • Language: English
  • Description: Nationallizenzen 142-005
  • Content: Accepted Version
  • Language: English
  • Description: Version 1
  • Content: Accepted Version
  • Language: English
  • Description: Version 2