On mean central limit theorems for stationary sequences
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 4, pp. 693-726.

In this paper, we give estimates of the minimal 𝕃 1 distance between the distribution of the normalized partial sum and the limiting gaussian distribution for stationary sequences satisfying projective criteria in the style of Gordin or weak dependence conditions.

Dans cet article, nous donnons des majorations de la distance minimale 𝕃 1 entre la loi de la somme normalisée et sa loi limite gaussienne pour des suites stationnaires satisfaisant des critères projectifs à la Gordin ou des conditions de dépendance faible.

DOI: 10.1214/07-AIHP117
Classification: 60F05
Keywords: mean central limit theorem, Wasserstein distance, minimal distance, martingale difference sequences, strong mixing, stationary sequences, weak dependence, rates of convergence, projective criteria
@article{AIHPB_2008__44_4_693_0,
     author = {Dedecker, J\'er\^ome and Rio, Emmanuel},
     title = {On mean central limit theorems for stationary sequences},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {693--726},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {4},
     year = {2008},
     doi = {10.1214/07-AIHP117},
     mrnumber = {2446294},
     zbl = {1187.60015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/07-AIHP117/}
}
TY  - JOUR
AU  - Dedecker, Jérôme
AU  - Rio, Emmanuel
TI  - On mean central limit theorems for stationary sequences
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2008
SP  - 693
EP  - 726
VL  - 44
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/07-AIHP117/
DO  - 10.1214/07-AIHP117
LA  - en
ID  - AIHPB_2008__44_4_693_0
ER  - 
%0 Journal Article
%A Dedecker, Jérôme
%A Rio, Emmanuel
%T On mean central limit theorems for stationary sequences
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2008
%P 693-726
%V 44
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/07-AIHP117/
%R 10.1214/07-AIHP117
%G en
%F AIHPB_2008__44_4_693_0
Dedecker, Jérôme; Rio, Emmanuel. On mean central limit theorems for stationary sequences. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 4, pp. 693-726. doi : 10.1214/07-AIHP117. http://www.numdam.org/articles/10.1214/07-AIHP117/

[1] R. P. Agnew. Global versions of the central limit theorem. Proc. Nat. Acad. Sci. U.S.A. 40 (1954) 800-804. | MR | Zbl

[2] H. Bergström. On the central limit theorem. Skand. Aktuarietidskr. 27 (1944) 139-153. | MR | Zbl

[3] E. Bolthausen. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw. Gebiete 54 (1980) 59-73. | MR | Zbl

[4] E. Bolthausen. Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (1982) 672-688. | MR | Zbl

[5] E. Bolthausen. The Berry-Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (1982) 283-289. | MR | Zbl

[6] J. Dedecker and F. Merlevède. Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044-1081. | MR | Zbl

[7] J. Dedecker and C. Prieur. New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 (2005) 203-236. | MR | Zbl

[8] J. Dedecker and E. Rio. On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. | Numdam | MR | Zbl

[9] Y. Derriennic and M. Lin. The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Related Fields 119 (2001) 508-528. | MR | Zbl

[10] R. Dudley. Real Analysis and Probability. Wadsworth Inc., Belmont, California, 1989. | MR | Zbl

[11] C.-G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm. 121 (1958) 1-30. | MR | Zbl

[12] M. Y. Fominykh. Properties of Riemann sums. Soviet Math. (Iz. VUZ) 29 (1985) 83-93. | MR | Zbl

[13] M. I. Gordin. The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR. 188 (1969) 739-741. | MR | Zbl

[14] M. I. Gordin. Abstracts of communication. In International Conference on Probability Theory, Vilnius, T.1: A-K, 1973.

[15] G. H. Hardy, J. E. Littlewood and G. Pólya. Inequalities. Cambridge University Press, 1952. | JFM | MR | Zbl

[16] I. A. Ibragimov. On asymptotic distribution of values of certain sums. Vestnik Leningrad. Univ. 15 (1960) 55-69. | MR | Zbl

[17] I. A. Ibragimov. The central limit theorem for sums of functions of independent variables and sums of type ∑f(2kt). Theory Probab. Appl. 12 (1967) 596-607. | MR | Zbl

[18] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Amsterdam, 1971. | MR | Zbl

[19] C. Jan. Vitesse de convergence dans le TCL pour des processus associés à des systèmes dynamiques et aux produits de matrices aléatoires. Thèse de l'université de Rennes 1, 2001.

[20] S. Le Borgne and F. Pène. Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques. Bull. Soc. Math. France 133 (2005) 395-417. | Numdam | MR | Zbl

[21] E. Nummelin. General Irreducible Markov Chains and non Negative Operators. Cambridge University Press, London, 1984. | MR | Zbl

[22] F. Pène. Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard. Ann. Appl. Probab. 15 (2005) 2331-2392. | MR | Zbl

[23] V. V. Petrov. Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford University Press, New York, 1995. | MR | Zbl

[24] E. Rio. About the Lindeberg method for strongly mixing sequences. ESAIM Probab. Statist. 1 (1995) 35-61. | Numdam | MR | Zbl

[25] E. Rio. Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 (1996) 255-282. | MR | Zbl

[26] E. Rio. Théorie asymptotique des processus aléatoires faiblement dépendants. Springer, Berlin, 2000. | MR | Zbl

[27] W. M. Schmidt. Diophantine Approximation. Springer, Berlin, 1980. | MR | Zbl

[28] I. Sunklodas. Distance in the L1 metric of the distribution of the sum of weakly dependent random variables from the normal distribution function. Litosvk. Mat. Sb. 22 (1982) 171-188. | MR | Zbl

[29] V. M. Zolotarev. On asymptotically best constants in refinements of mean limit theorems. Theory Probab. Appl. 9 (1964) 268-276. | MR | Zbl

Cited by Sources: