Comptes Rendus
no. 5
Probability Theory
An invariance principle for non-adapted processes
[Principe d'invariance pour processus non-adaptés]

Présenté par : Marc Yor

Jana Klicnarová1 ; Dalibor Volný2
1 Faculty of Economics, University of South Bohemia, Studentská 13, 370 05 České Budějovice, Czech Republic
2 Laboratoire de Mathématiques, Université de Rouen, Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France
Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 283-287.

Nous présentons un principe d'invariance conditionnel (par rapport à la tribu des ensembles invariants) pour une suite stationnaire non-adaptée de variables aléatoires. Il généralise le principe d'invariance de Wu et Woodroofe (2004, Corollary 3) en utilisant la méthode introduite par Volný (2006). A l'aide d'un exemple, nous montrons que la méthode ne donne pas une généralisation du principe d'invariance de Peligrad et Utev (2005).

We present an invariance principle for a non-adapted stationary sequence of random variables, conditional with respect to the σ-algebra of invariant sets. It is a generalization of an invariance principle of Wu and Woodroofe (2004, Corollary 3) using a method introduced by Volný (2006). An example shows that the method cannot be used directly for a generalization of the invariance principle of Peligrad and Utev (2005).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.05.009
Jana Klicnarová 1 ; Dalibor Volný 2

1 Faculty of Economics, University of South Bohemia, Studentská 13, 370 05 České Budějovice, Czech Republic
2 Laboratoire de Mathématiques, Université de Rouen, Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France 
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Jana Klicnarová; Dalibor Volný. An invariance principle for non-adapted processes. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 283-287. doi : 10.1016/j.crma.2007.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.05.009/

[1] J. Dedecker; F. Merlevède Necessary and sufficient conditions for the conditional central limit theorem, Ann. Probab., Volume 30 (2002), pp. 1044-1081

[2] J. Dedecker, F. Merlevède, D. Volný, On the weak invariance principle for non-adapted sequences under projective criteria J. Theor. Probab. (2007), in press

[3] M. Maxwell; M. Woodroofe Central limit theorems for additive functionals of Markov chains, Ann. Probab., Volume 28 (2000), pp. 713-724

[4] L. Ouchti, D. Volný, 2007, in preparation

[5] M. Peligrad; S. Utev A new maximal inequality and invariance principle for stationary sequences, Ann. Probab., Volume 33 (2005), pp. 798-815

[6] D. Volný On the invariance principle and functional law of iterated logarithm for nonergodic processes, Yokohama Math. J., Volume 35 (1987), pp. 137-141

[7] D. Volný Martingale approximation of non-adapted stochastic processes with nonlinear growth of variance (P. Bertail; P. Doukhan; P. Soulier, eds.), Dependence in Probability and Statistics Series, Lecture Notes in Statistics, vol. 187, Springer, 2006

[8] D. Volný, preprint, 2007

[9] W.B. Wu; M. Woodroofe Martingale approximations for sums of stationary processes, Ann. Prob., Volume 32 (2004) no. 2, pp. 1674-1690

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