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<α2. 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/nlogn 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<α2. 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/nlogn vers des distributions gaussiennes. Nos résultats montrent que ces lois ne sont précises que pour n d'une grandeur exponentielle.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.013
Börgers, Christoph 1; Greengard, Claude 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 = {B\"orgers, Christoph and Greengard, Claude},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.04.013/}
}
TY  - JOUR
AU  - Börgers, Christoph
AU  - Greengard, Claude
TI  - Slow convergence in generalized central limit theorems
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 679
EP  - 685
VL  - 356
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.04.013/
DO  - 10.1016/j.crma.2018.04.013
LA  - en
ID  - CRMATH_2018__356_6_679_0
ER  - 
%0 Journal Article
%A Börgers, Christoph
%A Greengard, Claude
%T Slow convergence in generalized central limit theorems
%J Comptes Rendus. Mathématique
%D 2018
%P 679-685
%V 356
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.04.013/
%R 10.1016/j.crma.2018.04.013
%G en
%F CRMATH_2018__356_6_679_0
Börgers, Christoph; Greengard, Claude. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.04.013/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: