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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2017-11-07
Accepted: 2018-04-12
Published online: 2018-04-19

DOI: 10.1016/j.crma.2018.04.013

Author's affiliations:

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

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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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