Probability theory
Slow convergence in generalized central limit theorems
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 679-685.

We study the central limit theorem in the non-normal domain of attraction to symmetric α-stable laws for $0<α≤2$. We show that for i.i.d. random variables $Xi$, the convergence rate in $L∞$ of both the densities and distributions of $∑inXi/(n1/αL(n))$ 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 $∑i=1nXi/nlog⁡n$ 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, $0<α≤2$. Nous montrons que, pour les suites $Xi$ i.i.d., les taux de convergence en $L∞$ des densités et des distributions de $∑inXi/(n1/αL(n))$ 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 $∑i=1nXi/nlog⁡n$ vers des distributions gaussiennes. Nos résultats montrent que ces lois ne sont précises que pour n d'une grandeur exponentielle.

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.013

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
@article{CRMATH_2018__356_6_679_0,
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},
}
TY  - JOUR
AU  - Christoph Börgers
AU  - Claude Greengard
TI  - Slow convergence in generalized central limit theorems
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 679
EP  - 685
VL  - 356
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2018.04.013
LA  - en
ID  - CRMATH_2018__356_6_679_0
ER  -
%0 Journal Article
%A Christoph Börgers
%A Claude Greengard
%T Slow convergence in generalized central limit theorems
%J Comptes Rendus. Mathématique
%D 2018
%P 679-685
%V 356
%N 6
%I Elsevier
%R 10.1016/j.crma.2018.04.013
%G en
%F CRMATH_2018__356_6_679_0
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/

[1] P. Bálint; S. Gouëzel Limit theorems in the stadium billiard, Commun. Math. Phys., Volume 263 (2006) no. 2, pp. 461-512

[2] C. Börgers; C. Greengard; E. Thomann The diffusion limit of free molecular flow in thin plane channels, SIAM J. Appl. Math., Volume 52 (1992) no. 4, pp. 1057-1075

[3] M.V. Boutsikas; M.V. Koutras Compound Poisson approximation for sums of dependent random variables (C.A. Charalambides; M.V. Koutras; N. Balakrishnan, eds.), Probability and Statistical Models with Applications: A Volume in Honour of Prof. T. Cacoullos, 2001, pp. 63-86

[4] G. Christoph; W. Wolf Convergence Theorems with a Stable Limit Law, Akademie Verlag, Berlin, 1992

[5] T. Chumley; R. Feres; H.-K. Zhang Diffusivity in multiple scattering systems, Trans. Amer. Math. Soc., Volume 368 (2016) no. 1, pp. 109-148

[6] G. Cristadoro; T. Gilbert; M. Lenci; D.P. Sanders Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards, Phys. Rev. E, Volume 90 (2014) no. 2

[7] C.P. Dettmann Diffusion in the Lorentz gas, Commun. Theor. Phys., Volume 62 (2014) no. 4, pp. 521-540

[8] S. Gouëzel Central limit theorem and stable laws for intermittent maps, Probab. Theory Relat. Fields, Volume 128 (2004) no. 1, pp. 82-122

[9] J. Jiménez Algebraic probability density tails in decaying isotropic two-dimensional turbulence, J. Fluid Mech., Volume 313 (1996), pp. 223-240

[10] A. Juozulynas; V. Paulauskas Some remarks on the rate of convergence to stable laws, Lith. Math. J., Volume 38 (1998) no. 4, pp. 335-347

[11] R. Kuske; J.B. Keller Rate of convergence to a stable law, SIAM J. Appl. Math., Volume 61 (2001) no. 4, pp. 1308-1323

[12] P. Nándori Recurrence properties of a special type of heavy-tailed random walk, J. Stat. Phys., Volume 142 (2011) no. 2, pp. 342-355

[13] J.P. Nolan Bibliography on stable distributions, processes and related topics, 2017 http://fs2.american.edu/jpnolan/www/stable/stable.html

[14] S.T. Rachev et al. The Methods of Distances in the Theory of Probability and Statistics, Springer Science & Business Media, 2013

[15] V.M. Zolotarev One-Dimensional Stable Distributions, Translations of Mathematical Monographs, vol. 65, American Mathematical Society, Providence, RI, USA, 1986 (in Russian)

Cited by Sources: