On mean central limit theorems for stationary sequences

Dedecker, Jérôme; Rio, Emmanuel

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 693-726.

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.

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.

Classification : 60F05
Mots clés : mean central limit theorem, Wasserstein distance, minimal distance, martingale difference sequences, strong mixing, stationary sequences, weak dependence, rates of convergence, projective criteria

[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