A quenched weak invariance principle
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 872-898.

Dans cet article, nous étudions le théorème central limite conditionnel presque sûr, ainsi que sa forme fonctionnelle, pour des suites stationnaires de variables aléatoires réelles satisfaisant une condition de type projectif. Nous donnons des applications de ces résultats aux processus fortement mélangeants ainsi qu'à des chaînes de Markov nonirréductibles. Les preuves sont essentiellement basées sur une approximation normale de suites doublement indexées de variables aléatoires de type martingale.

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and nonirreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, an approach which has interest in itself.

DOI : 10.1214/13-AIHP553
Classification : 60F05, 60F17, 60J05
Mots-clés : quenched central limit theorem, weak invariance principle, strong mixing, Markov chains 
@article{AIHPB_2014__50_3_872_0,
     author = {Dedecker, J\'er\^ome and Merlev\`ede, Florence and Peligrad, Magda},
     title = {A quenched weak invariance principle},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {872--898},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     doi = {10.1214/13-AIHP553},
     mrnumber = {3224292},
     zbl = {1304.60031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP553/}
}
TY  - JOUR
AU  - Dedecker, Jérôme
AU  - Merlevède, Florence
AU  - Peligrad, Magda
TI  - A quenched weak invariance principle
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 872
EP  - 898
VL  - 50
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP553/
DO  - 10.1214/13-AIHP553
LA  - en
ID  - AIHPB_2014__50_3_872_0
ER  - 
%0 Journal Article
%A Dedecker, Jérôme
%A Merlevède, Florence
%A Peligrad, Magda
%T A quenched weak invariance principle
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 872-898
%V 50
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP553/
%R 10.1214/13-AIHP553
%G en
%F AIHPB_2014__50_3_872_0
Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda. A quenched weak invariance principle. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 872-898. doi : 10.1214/13-AIHP553. http://archive.numdam.org/articles/10.1214/13-AIHP553/

[1] A. N. Borodin and I. A. Ibragimov. Limit Theorems for Functionals of Random Walks. Trudy Mat. Inst. Steklov. 195. Nauka, St. Petersburg, 1994. Transl. into English: Proc. Steklov Inst. Math. 195. Amer. Math. Soc., Providence, RI, 1995. | MR | Zbl

[2] R. C. Bradley. On quantiles and the central limit question for strongly mixing sequences. Dedicated to Murray Rosenblatt. J. Theoret. Probab. 10 (1997) 507-555. | MR | Zbl

[3] B. M. Brown. Martingale central limit theorems. Ann. Math. Statist. 42 (1971) 59-66. | MR | Zbl

[4] X. Chen. Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139 (1999) xiv+203. | MR | Zbl

[5] C. Cuny. Pointwise ergodic theorems with rate and application to limit theorems for stationary processes. Stoch. Dyn. 11 (2011) 135-155. | MR | Zbl

[6] C. Cuny and F. Merlevède. On martingale approximations and the quenched weak invariance principle. Ann. Probab. 42 (2014) 760-793. | MR

[7] C. Cuny and M. Peligrad. Central limit theorem started at a point for additive functional of reversible Markov Chains. J. Theoret. Probab. 25 (2012) 171-188. | MR | Zbl

[8] C. Cuny and D. Volný. A quenched invariance principle for stationary processes. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 107-115. | MR | Zbl

[9] J. Dedecker, S. Gouëzel and F. Merlevède. Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 796-821. | Numdam | MR | Zbl

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

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

[12] 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

[13] Y. Derriennic and M. Lin. The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 (2003) 73-76. | MR | Zbl

[14] P. Doukhan, P. Massart and E. Rio. The functional central limit theorem for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Stat. 30 (1994) 63-82. | Numdam | MR | Zbl

[15] O. Durieu. Independence of four projective criteria for the weak invariance principle. ALEA Lat. Am. J. Probab. Math. Stat. 5 (2009) 21-26. | MR | Zbl

[16] O. Durieu and D. Volný. Comparison between criteria leading to the weak invariance principle. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 324-340. | Numdam | MR | Zbl

[17] C. G. Esseen and S. Janson. On moment conditions for normed sums of independent variables and martingale differences. Stochastic Process. Appl. 19 (1985) 173-182. | MR | Zbl

[18] M. I. Gordin. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174-1176. | MR | Zbl

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

[20] M. I. Gordin and B. A. Lifsic. The central limit theorem for stationary Markov processes. Soviet Math. Dokl. 19 (1978) 392-394. | MR | Zbl

[21] S. Gouëzel. Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 (2004) 82-122. | MR | Zbl

[22] C. C. Heyde and B. M. Brown. On the departure from normality of a certain class of martingales. Ann. Math. Statist. 41 (1970) 2161-2165. | MR | Zbl

[23] U. Krengel. Ergodic Theorems. de Gruyter Studies in Mathematics 6. de Gruyter, Berlin, 1985. | MR | Zbl

[24] M. Maxwell and M. Woodroofe. Central limit theorem for additive fonctionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR | Zbl

[25] F. Merlevède, M. Peligrad and S. Utev. Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 (2006) 1-36. | MR | Zbl

[26] F. Merlevède, C. Peligrad and M. Peligrad. Almost sure invariance principles via martingale approximation. Stochastic Process. Appl. 122 (2012) 170-190. | MR | Zbl

[27] F. Merlevède and E. Rio. Strong approximation of partial sums under dependence conditions with application to dynamical systems. Stochastic Process. Appl. 122 (2012) 386-417. | MR | Zbl

[28] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag, London, 1993. | MR | Zbl

[29] M. Peligrad and S. Utev. Central limit theorem for stationary linear processes. Ann. Probab. 34 (2006) 1608-1622. | MR | Zbl

[30] Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980) 189-197. | MR | Zbl

[31] M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. | MR | Zbl

[32] Ja. G. Sinaĭ. A weak isomorphism of transformations with invariant measure. Dokl. Akad. Nauk SSSR 147 (1962) 797-800. | MR | Zbl

[33] D. Volný and P. Samek. On the invariance principle and the law of iterated logarithm for stationary processes. In Mathematical Physics and Stochastic Analysis 424-438. World Sci. Publishing, River Edge, 2000. | MR | Zbl

[34] D. Volný and M. Woodroofe. An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process. In Dependence in Analysis, Probability and Number Theory (The Phillipp memorial volume) 317-323. Kendrick Press, Heber City, UT, 2010. | MR | Zbl

[35] D. Volný and M. Woodroofe. Quenched central limit theorems for sums of stationary processes. Preprint, 2010. Available at arXiv:1006.1795. | MR | Zbl

[36] W.-B. Wu and M. Woodroofe. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004) 1674-1690. | MR | Zbl

[37] O. Zhao and M. Woodroofe. Law of the iterated logarithm for stationary processes. Ann. Probab. 36 (2008) 127-142. | MR | Zbl

Cité par Sources :