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Mod-Gaussian convergence: new limit theorems in probability and number theory
-
Jean Jacod
,
Emmanuel Kowalski
Published/Copyright:
April 14, 2010
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.
Keywords.: Limit theorems; random matrices; characteristic polynomial; infinitely divisible distributions; zeta and L-functions; Katz–Sarnak philosophy; Erdős–Kac Theorem
Received: 2008-07-30
Revised: 2009-11-02
Published Online: 2010-04-14
Published in Print: 2011-July
© de Gruyter 2011
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Keywords for this article
Limit theorems;
random matrices;
characteristic polynomial;
infinitely divisible distributions;
zeta and L-functions;
Katz–Sarnak philosophy;
Erdős–Kac Theorem
Articles in the same Issue
- The sh-Lie algebra perturbation lemma
- Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains
- Geography of non-formal symplectic and contact manifolds
- Hp → Hp boundedness implies Hp → Lp boundedness
- Global regularity for the minimal surface equation in Minkowskian geometry
- On elementarily κ-homogeneous unary structures
- Kernels, regularity and unipotent radicals in linear algebraic monoids
- Mod-Gaussian convergence: new limit theorems in probability and number theory
- Uniform large deviation for pinned hyperbolic Brownian motion
