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Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus


Chhaibi, Reda; Delbaen, Freddy; Méliot, Pierre-Loïc; Nikeghbali, Ashkan (2020). Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus. Annales de l'Institut Fourier, 70(3):1115-1197.

Abstract

In this paper, we relate the framework of mod-φ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is the one of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents of the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in permutations, number of prime divisors of a random integer, number of irreducible factors of a random polynomial, etc. Our approach allows us to deal with approximations in higher dimensions as well. In this setting, we bring out the influence of the correlations between the components of the random vectors in our asymptotic formulas.

Abstract

In this paper, we relate the framework of mod-φ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is the one of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents of the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in permutations, number of prime divisors of a random integer, number of irreducible factors of a random polynomial, etc. Our approach allows us to deal with approximations in higher dimensions as well. In this setting, we bring out the influence of the correlations between the components of the random vectors in our asymptotic formulas.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Banking and Finance
07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Algebra and Number Theory
Physical Sciences > Geometry and Topology
Scope:Discipline-based scholarship (basic research)
Language:English
Date:26 June 2020
Deposited On:12 Oct 2020 12:35
Last Modified:23 Apr 2024 01:44
Publisher:Association des Annales de l'Institut Fourier
ISSN:0373-0956
OA Status:Gold
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.5802/aif.3332
Other Identification Number:merlin-id:19436
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution-NoDerivs 3.0 Unported (CC BY-ND 3.0)