On mean central limit theorems for stationary sequences
Annales de l'I.H.P. Probabilités et statistiques (2008)
- Volume: 44, Issue: 4, page 693-726
- ISSN: 0246-0203
Access Full Article
topAbstract
topHow to cite
topDedecker, Jérôme, and Rio, Emmanuel. "On mean central limit theorems for stationary sequences." Annales de l'I.H.P. Probabilités et statistiques 44.4 (2008): 693-726. <http://eudml.org/doc/77988>.
@article{Dedecker2008,
abstract = {In this paper, we give estimates of the minimal $\{\mathbb \{L\}\}^\{1\}$ 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.},
author = {Dedecker, Jérôme, Rio, Emmanuel},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {mean central limit theorem; Wasserstein distance; minimal distance; martingale difference sequences; strong mixing; stationary sequences; weak dependence; rates of convergence; projective criteria},
language = {eng},
number = {4},
pages = {693-726},
publisher = {Gauthier-Villars},
title = {On mean central limit theorems for stationary sequences},
url = {http://eudml.org/doc/77988},
volume = {44},
year = {2008},
}
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
PB - Gauthier-Villars
VL - 44
IS - 4
SP - 693
EP - 726
AB - In this paper, we give estimates of the minimal ${\mathbb {L}}^{1}$ 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.
LA - eng
KW - mean central limit theorem; Wasserstein distance; minimal distance; martingale difference sequences; strong mixing; stationary sequences; weak dependence; rates of convergence; projective criteria
UR - http://eudml.org/doc/77988
ER -
References
top- [1] R. P. Agnew. Global versions of the central limit theorem. Proc. Nat. Acad. Sci. U.S.A. 40 (1954) 800–804. Zbl0055.36703MR64342
- [2] H. Bergström. On the central limit theorem. Skand. Aktuarietidskr. 27 (1944) 139–153. Zbl0060.28707MR15703
- [3] E. Bolthausen. The Berry–Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw. Gebiete 54 (1980) 59–73. Zbl0431.60019MR595481
- [4] E. Bolthausen. Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (1982) 672–688. Zbl0494.60020MR659537
- [5] E. Bolthausen. The Berry–Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (1982) 283–289. Zbl0476.60022MR664418
- [6] J. Dedecker and F. Merlevède. Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044–1081. Zbl1015.60016MR1920101
- [7] J. Dedecker and C. Prieur. New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 (2005) 203–236. Zbl1061.62058MR2199291
- [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. Zbl0949.60049MR1743095
- [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. Zbl0974.60017MR1826405
- [10] R. Dudley. Real Analysis and Probability. Wadsworth Inc., Belmont, California, 1989. Zbl0686.60001MR982264
- [11] C.-G. Esseen. On mean central limit theorems. Kungl. Tekn. Högsk. Handl. Stockholm. 121 (1958) 1–30. Zbl0081.35202MR97111
- [12] M. Y. Fominykh. Properties of Riemann sums. Soviet Math. (Iz. VUZ) 29 (1985) 83–93. Zbl0603.41012MR796592
- [13] M. I. Gordin. The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR. 188 (1969) 739–741. Zbl0212.50005MR251785
- [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. Zbl0047.05302MR46395JFM60.0169.01
- [16] I. A. Ibragimov. On asymptotic distribution of values of certain sums. Vestnik Leningrad. Univ. 15 (1960) 55–69. Zbl0201.50701MR120679
- [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. Zbl0217.49803MR226711
- [18] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Amsterdam, 1971. Zbl0219.60027MR322926
- [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. Zbl1090.37018MR2169824
- [21] E. Nummelin. General Irreducible Markov Chains and non Negative Operators. Cambridge University Press, London, 1984. Zbl0551.60066MR776608
- [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. Zbl1097.37030MR2187297
- [23] V. V. Petrov. Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford University Press, New York, 1995. Zbl0826.60001MR1353441
- [24] E. Rio. About the Lindeberg method for strongly mixing sequences. ESAIM Probab. Statist. 1 (1995) 35–61. Zbl0869.60021MR1382517
- [25] E. Rio. Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 (1996) 255–282. Zbl0838.60017MR1373378
- [26] E. Rio. Théorie asymptotique des processus aléatoires faiblement dépendants. Springer, Berlin, 2000. Zbl0944.60008MR2117923
- [27] W. M. Schmidt. Diophantine Approximation. Springer, Berlin, 1980. Zbl0421.10019MR568710
- [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. Zbl0495.60039MR659030
- [29] V. M. Zolotarev. On asymptotically best constants in refinements of mean limit theorems. Theory Probab. Appl. 9 (1964) 268–276. Zbl0137.12101MR163338
Citations in EuDML Documents
top- J. Dedecker, S. Gouëzel, F. Merlevède, Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains
- Jérôme Dedecker, Florence Merlevède, Magda Peligrad, Sergey Utev, Moderate deviations for stationary sequences of bounded random variables
- Loïc Hervé, Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques
- Loïc Hervé, Françoise Pène, The Nagaev-Guivarc’h method via the Keller-Liverani theorem
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.