Soit T un opérateur de Dunford-Schwartz sur un espace de probabilité (Ω, μ). Pour f∈Lp(μ), p>1, nous obtenons des théorèmes ergodiques du type (1/n1/p)∑k=1nTkf→0 μ-p.s. sous des conditions portant sur la croissance de ‖∑k=1nTkf‖p. Lorsque T est induit par une transformation préservant la mesure et que p=2, nous obtenons de meilleurs résultats. Ces derniers sont alors utilisés pour obtenir le théorème central limite «quenched» pour les sommes partielles associées aux fonctionnelles de chaînes de Markov stationnaires et ergodiques. Nous améliorons ainsi des résultats antérieurs de Derriennic-Lin et Wu-Woodroofe.
Let T be Dunford-Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic-Lin and Wu-Woodroofe.
Mots clés : ergodic theorems with rates, central limit theorem for Markov chains, Dunford-Schwartz operators, probability preserving transformations
@article{AIHPB_2009__45_3_710_0, author = {Cuny, Christophe and Lin, Michael}, title = {Pointwise ergodic theorems with rate and application to the {CLT} for {Markov} chains}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {710--733}, publisher = {Gauthier-Villars}, volume = {45}, number = {3}, year = {2009}, doi = {10.1214/08-AIHP180}, mrnumber = {2548500}, zbl = {1186.37013}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP180/} }
TY - JOUR AU - Cuny, Christophe AU - Lin, Michael TI - Pointwise ergodic theorems with rate and application to the CLT for Markov chains JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 710 EP - 733 VL - 45 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP180/ DO - 10.1214/08-AIHP180 LA - en ID - AIHPB_2009__45_3_710_0 ER -
%0 Journal Article %A Cuny, Christophe %A Lin, Michael %T Pointwise ergodic theorems with rate and application to the CLT for Markov chains %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 710-733 %V 45 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP180/ %R 10.1214/08-AIHP180 %G en %F AIHPB_2009__45_3_710_0
Cuny, Christophe; Lin, Michael. Pointwise ergodic theorems with rate and application to the CLT for Markov chains. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 710-733. doi : 10.1214/08-AIHP180. http://www.numdam.org/articles/10.1214/08-AIHP180/
[1] On the one-sided ergodic Hilbert transform. Contemp. Math. 430 (2007) 20-39. | MR | Zbl
and .[2] Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory. Israel J. Math. 148 (2005) 41-86. | MR | Zbl
and .[3] On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. | Numdam | MR | Zbl
and .[4] Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the “central limit theorem.” Discrete Contin. Dyn. Syst. 15 (2006) 143-158. | MR | Zbl
.[5] Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123 (2001) 93-130. | MR | Zbl
and .[6] 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
and .[7] The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 (2003) 73-76. | MR | Zbl
and .[8] The central limit theorem for random walks on orbits of probability preserving transformations. Contemp. Math. 444 (2007) 31-51. | MR | Zbl
and .[9] Linear Operators, Part I. Wiley, New York. 1958. | MR | Zbl
and .[10] Linear Operators, Part II. Wiley, New York, 1963. | MR | Zbl
and .[11] An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl
.[12] On the dependence of the convergence rate in the SLLN for stationary processes on the rate of decay of correlation function. Theory Probab. Appl. 26 (1981) 706-720. | MR | Zbl
.[13] Spectral criteria for the existence of generalized ergodic transformations (in Russian). Teor. Veroyatnost. i Primenen. 41(2) (1996) 251-271. (Translation in Theory Probab. Appl. 41 (1996) 247-264 (1997).) | MR | Zbl
.[14] A central limit theorem for Markov process. Soviet Math. Doklady 19 (1978) 392-394. | Zbl
and .[15] A remark about a Markov process with normal transition operator. Proc. Third Vilnius Conf. Probab. Statist. 147-148. Akad. Nauk Litovsk., Vilnius, 1981 (in Russian).
and .[16] The central limit theorem for Markov processes with normal transition operator, and a strong form of the central limit theorem. In Limit Theorems for Functionals of Random Walks Sections IV.7 and IV.8. A. Borodin and I. Ibragimov (Eds). Proc. Steklov Inst. Math. 195, 1994. (English translation Amer. Math. Soc., Providence, RI, 1995.)
and .[17] The rate of convergence in ergodic theorems. Russian Math. Surveys 51 (1996) 653-703. | MR | Zbl
.[18] Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl
and .[19] Ergodic Theorems. De Gruyter, Berlin, 1985. | MR | Zbl
.[20] Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR | Zbl
and .[21] Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35 (1976) 299-314. | MR | Zbl
.[22] A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798-815. | MR | Zbl
and .[23] An almost sure invariance principle for additive functionals of Markov chains. Statist. Probab. Lett. 78 (2008) 854-860. | MR | Zbl
and . and . Leçons D'analyse Fonctionnelle, 3rd edition. Akadémiai Kiadó, Budapest, 1955. |[25] Uniform bounds under increment conditions. Trans. Amer. Math. Soc. 358 (2006) 911-936. | MR | Zbl
.[26] Strong invariance principles for dependent random variables. Ann. Probab. 35 (2007) 2294-2320. | MR | Zbl
.[27] Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004) 1674-1690. | MR | Zbl
and .[28] Laws of the iterated logarithm for stationary processes. Ann. Probab. 36 (2008) 127-142. | MR | Zbl
and .[29] Trigonometric Series, corrected 2nd edition. Cambridge Univ. Press, Cambridge, UK, 1969. | Zbl
.Cité par Sources :