Slow convergence in generalized central limit theorems

Christoph Börgers; Claude Greengard

doi:10.1016/j.crma.2018.04.013

Probability theory

Slow convergence in generalized central limit theorems
[Convergence lente dans les théorèmes centraux limites généralisés]

Présenté par : Jean-François Le Gall

Christoph Börgers ¹ ; Claude Greengard ²

¹ Department of Mathematics, Tufts University, Medford, MA, 02155, United States
² Courant Institute of Mathematical Sciences, New York University and Foss Hill Partners, P.O. Box 938, Chappaqua, NY 10514, United States

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 679-685.

Résumé
Abstract

Nous étudions le théorème central limite dans le domaine d'attraction non normal, vers des limites symétriques et α-stables, $0 < α \leq 2$ . Nous montrons que, pour les suites $X_{i}$ i.i.d., les taux de convergence en $L^{\infty}$ des densités et des distributions de $\sum_{i}^{n} X_{i} / (n^{1 / α} L (n))$ sont au plus logarithmiques si L est une fonction non triviale de variation lente. Plusieurs lois physiques asymptotiques sont basées sur la convergence des suites $\sum_{i = 1}^{n} X_{i} / \sqrt{n \log n}$ vers des distributions gaussiennes. Nos résultats montrent que ces lois ne sont précises que pour n d'une grandeur exponentielle.

We study the central limit theorem in the non-normal domain of attraction to symmetric α-stable laws for $0 < α \leq 2$ . We show that for i.i.d. random variables $X_{i}$ , the convergence rate in $L^{\infty}$ of both the densities and distributions of ${\sum_{i}}^{n} X_{i} / (n^{1 / α} L (n))$ is at best logarithmic if L is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using convergence of ${\sum_{i = 1}}^{n} X_{i} / \sqrt{n \log n}$ to Gaussian distributions. Our result implies that such asymptotic laws are accurate only for exponentially large n.

Reçu le : 2017-11-07
Accepté le : 2018-04-12
Publié le : 2018-04-19

DOI : 10.1016/j.crma.2018.04.013

Affiliations des auteurs :

Christoph Börgers ¹ ; Claude Greengard ²

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     author = {Christoph B\"orgers and Claude Greengard},
     title = {Slow convergence in generalized central limit theorems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {679--685},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.013},
     language = {en},
}

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Christoph Börgers; Claude Greengard. Slow convergence in generalized central limit theorems. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 679-685. doi : 10.1016/j.crma.2018.04.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.013/

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