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

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.

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.

DOI : https://doi.org/10.1214/13-AIHP553
Classification:  60F05,  60F17,  60J05
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {872-898},
     doi = {10.1214/13-AIHP553},
     zbl = {1304.60031},
     mrnumber = {3224292},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 50 (2014) no. 3, pp. 872-898. doi : 10.1214/13-AIHP553. http://www.numdam.org/item/AIHPB_2014__50_3_872_0/

[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 1368394 | Zbl 0855.60001

[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 1455156 | Zbl 0887.60028

[3] B. M. Brown. Martingale central limit theorems. Ann. Math. Statist. 42 (1971) 59-66. | MR 290428 | Zbl 0218.60048

[4] X. Chen. Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139 (1999) xiv+203. | MR 1491814 | Zbl 0952.60014

[5] C. Cuny. Pointwise ergodic theorems with rate and application to limit theorems for stationary processes. Stoch. Dyn. 11 (2011) 135-155. | MR 2771346 | Zbl 1210.60031

[6] C. Cuny and F. Merlevède. On martingale approximations and the quenched weak invariance principle. Ann. Probab. 42 (2014) 760-793. | MR 3178473 | Zbl pre06288293

[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 2886384 | Zbl 1247.60031

[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 3083921 | Zbl 1277.60056

[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 2682267 | Zbl 1206.60032

[10] J. Dedecker and F. Merlevède. Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044-1081. | MR 1920101 | Zbl 1015.60016

[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 1743095 | Zbl 0949.60049

[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 1826405 | Zbl 0974.60017

[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 1952457 | Zbl 1012.60028

[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 1262892 | Zbl 0790.60037

[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 2475604 | Zbl 1169.60005

[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 2446326 | Zbl 1182.60010

[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 780729 | Zbl 0554.60050

[18] M. I. Gordin. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174-1176. | MR 251785 | Zbl 0212.50005

[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 501277 | Zbl 0395.60057

[21] S. Gouëzel. Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 (2004) 82-122. | MR 2027296 | Zbl 1038.37007

[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 293702 | Zbl 0225.60026

[23] U. Krengel. Ergodic Theorems. de Gruyter Studies in Mathematics 6. de Gruyter, Berlin, 1985. | MR 797411 | Zbl 0575.28009

[24] M. Maxwell and M. Woodroofe. Central limit theorem for additive fonctionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR 1782272 | Zbl 1044.60014

[25] F. Merlevède, M. Peligrad and S. Utev. Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 (2006) 1-36. | MR 2206313 | Zbl 1189.60078

[26] F. Merlevède, C. Peligrad and M. Peligrad. Almost sure invariance principles via martingale approximation. Stochastic Process. Appl. 122 (2012) 170-190. | MR 2860446 | Zbl 1230.60029

[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 2860454 | Zbl 1230.60034

[28] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag, London, 1993. | MR 1287609 | Zbl 0925.60001

[29] M. Peligrad and S. Utev. Central limit theorem for stationary linear processes. Ann. Probab. 34 (2006) 1608-1622. | MR 2257658 | Zbl 1101.60014

[30] Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980) 189-197. | MR 576270 | Zbl 0578.76059

[31] M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. | MR 74711 | Zbl 0070.13804

[32] Ja. G. Sinaĭ. A weak isomorphism of transformations with invariant measure. Dokl. Akad. Nauk SSSR 147 (1962) 797-800. | MR 161960 | Zbl 0205.13501

[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 1893125 | Zbl 0974.60013

[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 2731055 | Zbl 1213.60051

[35] D. Volný and M. Woodroofe. Quenched central limit theorems for sums of stationary processes. Preprint, 2010. Available at arXiv:1006.1795. | MR 3157895 | Zbl 1290.60027

[36] W.-B. Wu and M. Woodroofe. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004) 1674-1690. | MR 2060314 | Zbl 1057.60022

[37] O. Zhao and M. Woodroofe. Law of the iterated logarithm for stationary processes. Ann. Probab. 36 (2008) 127-142. | MR 2370600 | Zbl 1130.60039