Comparison between criteria leading to the weak invariance principle

Durieu, Olivier; Volný, Dalibor

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 324-340.

Le but de cet article est de comparer différents critères conduisant au théoreme limite centrale et au principe d'invariance faible. Ces critères sont la décomposition martingale-cobord développée par Gordin dans Dokl. Akad. Nauk SSSR 188 (1969), le critère projectif introduit par Dedecker dans Probab. Theory Related Fields 110 (1998), par la suite amélioré par Dedecker et Rio dans Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) et la condition introduite par Maxwell et Woodroofe dans Ann. Probab. 28 (2000), plus tard améliorée par Peligrad et Utev dans Ann. Probab. 33 (2005). On montre que dans tout système dynamique ergodique d’entropie strictement positive, si l’on considère deux de ces critères, on peut trouver une fonction dans ${𝕃}^2$ vérifiant le premier mais pas le deuxième.

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in ${𝕃}^2$ satisfying the first but not the second.

[1] P. Billingsley. The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12 (1961) 788-792. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[3] Y. S. Chow and H. Teicher. Probability Theory, 3rd edition. Springer, New York, 1997. | MR | Zbl

[4] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ. Ergodic Theory. Springer, New York, 1982. (Translated from the Russian by A. B. Sosinskiĭ.) | MR | Zbl

[5] J. Dedecker. A central limit theorem for stationary random fields. Probab. Theory Related Fields 110 (1998) 397-426. | MR | Zbl

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

[7] A. Del Junco and J. Rosenblatt. Counterexamples in ergodic theory and number theory. Math. Ann. 245 (1979) 185-197. | MR | Zbl

[8] M. I. Gordin. The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739-741. | MR | Zbl

[9] A. Gut. Probability: A Graduate Course. Springer, New York, 2005. | MR | Zbl

[10] P. Hall and C. C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980. | MR | Zbl

[11] C. C. Heyde. On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12 (1975) 1-8. | MR | Zbl

[12] I. A. Ibragimov. A central limit theorem for a class of dependent random variables. Teor. Verojatnost. i Primenen. 8 (1963) 89-94. | MR | Zbl

[13] S. Le Borgne. Limit theorems for non-hyperbolic automorphisms of the torus. Israel J. Math. 109 (1999) 61-73. | MR | Zbl

[14] C. Liverani. Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995) 56-75. Pitman Res. Notes Math. Ser. 362. Longman, Harlow, 1996. | MR | Zbl

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

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

[17] D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Advances in Math. 4 (1970) 337-352. | MR | Zbl

[18] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798-815. | MR | Zbl

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

[20] D. Volný. Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Process. Appl. 44 (1993) 41-74. | MR | Zbl

[21] D. Volný and P. Samek. On the invariance principle and the law of iterated logarithm for stationary processes. In Mathematical Physics and Stochastic Analysis (Lisbon, 1998) 424-438. World Sci. Publishing, River Edge, NJ, 2000. | MR | Zbl