×

On mean central limit theorems for stationary sequences. (English) Zbl 1187.60015

The authors investigate the normal approximation of a sum of dependent real-valued random variables and use Lindeberg’s method to prove bounds for the \(L^1\) distance between the distribution functions. In particular, they consider stationary random variables satisfying either projective criteria in the style of M. I. Gordin [Abstracts of communication. in: International Conference on Probability Theory, Vilnius, T.1: A-K (1973)] or weak dependence conditions.
Reviewer: Bero Roos (Trier)

MSC:

60F05 Central limit and other weak theorems
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] R. P. Agnew. Global versions of the central limit theorem. Proc. Nat. Acad. Sci. U.S.A. 40 (1954) 800-804. JSTOR: · Zbl 0055.36703 · doi:10.1073/pnas.40.9.800
[2] H. Bergström. On the central limit theorem. Skand. Aktuarietidskr. 27 (1944) 139-153. · Zbl 0060.28707
[3] E. Bolthausen. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw. Gebiete 54 (1980) 59-73. · Zbl 0431.60019 · doi:10.1007/BF00535354
[4] E. Bolthausen. Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (1982) 672-688. · Zbl 0494.60020 · doi:10.1214/aop/1176993776
[5] E. Bolthausen. The Berry-Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (1982) 283-289. · Zbl 0476.60022 · doi:10.1007/BF00535716
[6] J. Dedecker and F. Merlevède. Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044-1081. · Zbl 1015.60016 · doi:10.1214/aop/1029867121
[7] J. Dedecker and C. Prieur. New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 (2005) 203-236. · Zbl 1061.62058 · doi:10.1007/s00440-004-0394-3
[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. · Zbl 0949.60049 · doi:10.1016/S0246-0203(00)00111-4
[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. · Zbl 0974.60017 · doi:10.1007/s004400000113
[10] R. Dudley. Real Analysis and Probability . Wadsworth Inc., Belmont, California, 1989. · Zbl 0686.60001
[11] C.-G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm . 121 (1958) 1-30. · Zbl 0081.35202
[12] M. Y. Fominykh. Properties of Riemann sums. Soviet Math. (Iz. VUZ) 29 (1985) 83-93. · Zbl 0603.41012
[13] M. I. Gordin. The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR . 188 (1969) 739-741. · Zbl 0212.50005
[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.
[16] I. A. Ibragimov. On asymptotic distribution of values of certain sums. Vestnik Leningrad. Univ. 15 (1960) 55-69. · Zbl 0202.46705
[17] I. A. Ibragimov. The central limit theorem for sums of functions of independent variables and sums of type \sum f (2 k t ). Theory Probab. Appl. 12 (1967) 596-607. · Zbl 0217.49803 · doi:10.1137/1112075
[18] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables . Wolters-Noordhoff, Amsterdam, 1971. · Zbl 0219.60027
[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. · Zbl 1090.37018
[21] E. Nummelin. General Irreducible Markov Chains and non Negative Operators . Cambridge University Press, London, 1984. · Zbl 0551.60066
[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. · Zbl 1097.37030 · doi:10.1214/105051605000000476
[23] V. V. Petrov. Limit Theorems of Probability Theory. Sequences of Independent Random Variables . Oxford University Press, New York, 1995. · Zbl 0826.60001
[24] E. Rio. About the Lindeberg method for strongly mixing sequences. ESAIM Probab. Statist. 1 (1995) 35-61. · Zbl 0869.60021 · doi:10.1051/ps:1997102
[25] E. Rio. Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 (1996) 255-282. · Zbl 0838.60017 · doi:10.1007/BF01247840
[26] E. Rio. Théorie asymptotique des processus aléatoires faiblement dépendants . Springer, Berlin, 2000.
[27] W. M. Schmidt. Diophantine Approximation . Springer, Berlin, 1980. · Zbl 0421.10019
[28] I. Sunklodas. Distance in the L 1 metric of the distribution of the sum of weakly dependent random variables from the normal distribution function. Litosvk. Mat. Sb. 22 (1982) 171-188. · Zbl 0495.60039
[29] V. M. Zolotarev. On asymptotically best constants in refinements of mean limit theorems. Theory Probab. Appl. 9 (1964) 268-276. · Zbl 0137.12101 · doi:10.1137/1109039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.