Mod-φ convergence: Approximation of discrete measures and harmonic analysis on the torus
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1115-1197.

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.

Dans cet article, nous relions la théorie de la convergence mod-φ à la construction de schémas d’approximation pour des variables aléatoires à valeurs dans des réseaux. Le point de vue adopté est celui de l’analyse de Fourier dans l’algèbre de Wiener ; il permet le calcul d’équivalents asymptotiques des distances locales, de Kolmogorov et en variation totale. En utilisant des mesures signées au lieu de mesures de probabilités, nous construisons des approximations de distributions discrètes meilleures que l’approximation standard poissonnienne. Cette théorie s’applique à divers exemples issus de la combinatoire et de la théorie des nombres : nombre de cycles dans des permutations, nombre de diviseurs premiers d’un entier aléatoire, nombre de facteurs irréductibles d’un polynôme aléatoire, etc. Notre approche permet également des approximations en dimension supérieure. Dans ce cadre, nous mettons en évidence l’influence sur nos formules asymptotiques des corrélations entre les composantes des vecteurs aléatoires.

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Accepted:
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DOI: 10.5802/aif.3332
Classification: 62E17,  62E20,  60E10,  11N37,  11N45
Keywords: Mod-φ convergence, Wiener algebra, Lattice distributions, Approximation of random variables
Chhaibi, Reda 1; Delbaen, Freddy 2; Méliot, Pierre-Loïc 3; Nikeghbali, Ashkan 2

1 Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse cedex 9 (France)
2 ETH Zürich, Department of Mathematics and Universität Zürich, Institut für Mathematik Winterthurerstrasse 190 CH-8057 Zürich (Switzerland)
3 Laboratoire de Mathématiques, Bâtiment 307 Faculté des sciences d’Orsay Université Paris-Sud 91400 Orsay (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Chhaibi, Reda; Delbaen, Freddy; Méliot, Pierre-Loïc; Nikeghbali, Ashkan. Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1115-1197. doi : 10.5802/aif.3332. https://aif.centre-mersenne.org/articles/10.5802/aif.3332/

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