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Dependent Lindeberg central limit theorem and some applications

Published online by Cambridge University Press:  23 January 2008

Jean-Marc Bardet ,
Paul Doukhan ,
Gabriel Lang  and
Nicolas Ragache
Jean-Marc Bardet
Affiliation:
Samos-Matisse-CES, Université Panthéon-Sorbonne, 90 rue de Tolbiac, 75013 Paris, France.
Paul Doukhan
Affiliation:
Samos-Matisse-CES, Université Panthéon-Sorbonne, 90 rue de Tolbiac, 75013 Paris, France. LS-CREST, Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, France.
Gabriel Lang
Affiliation:
AgroParisTech, UMR MIA 518 (AgroParisTech-INRA), 75005 Paris, France.
Nicolas Ragache
Affiliation:
LS-CREST, Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, France.
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Abstract

In this paper, a very useful lemma (in two versions) is proved: itsimplifies notably the essential step to establish a Lindebergcentral limit theorem for dependent processes. Then, applying thislemma to weakly dependent processes introduced in Doukhan andLouhichi (1999), a new central limit theorem is obtained forsample mean or kernel density estimator. Moreover, by using thesubsampling, extensions under weaker assumptions of these centrallimit theorems are provided. All the usual causal or non causaltime series: Gaussian, associated, linear, ARCH(∞),bilinear, Volterra processes, ..., enter this frame.

Keywords

Central limit theoremLindeberg methodweak dependencekernel density estimationsubsampling
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 154 - 172
Copyright
© EDP Sciences, SMAI, 2008

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References

Andrews, D., Non strong mixing autoregressive processes. J. Appl. Probab. 21 (1984) 930–934. CrossRef
P. Billingsley, Convergence of Probability Measures. Wiley, New-York (1968).
Bulinski, A.V. and Shashkin, A.P., Rates in the central limit theorem for weakly dependent random variables. J. Math. Sci. 122 (2004) 3343–3358. CrossRef
Bulinski, A.V. and Shashkin, A.P., Strong Invariance Principle for Dependent Multi-indexed Random Variables. Doklady Mathematics 72 (2005) 503–506.
Coulon-Prieur, C. and Doukhan, P., A triangular central limit theorem under a new weak dependence condition. Stat. Prob. Letters 47 (2000) 61–68. CrossRef
P. Doukhan, Mixing: Properties and Examples. Lect. Notes Statis. 85 (1994).
P. Doukhan, Models inequalities and limit theorems for stationary sequences, in Theory and applications of long range dependence, Doukhan et al. Ed., Birkhäuser (2003) 43–101.
Doukhan, P. and Lang, G., Rates in the empirical central limit theorem for stationary weakly dependent random fields. Stat. Inference Stoch. Process. 5 (2002) 199–228. CrossRef
Doukhan, P. and Louhichi, S., A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 84 (1999) 313–342. CrossRef
Doukhan, P., Madre, H. and Rosenbaum, M., Weak dependence for infinite ARCH-type bilinear models. Statistics 41 (2007) 31–45. CrossRef
P. Doukhan, G. Teyssiere and P. Winant, Vector valued ARCH(∞) processes, in Dependence in Probability and Statistics, P. Bertail, P. Doukhan and P. Soulier Eds. Lecture Notes in Statistics, Springer, New York (2006).
Doukhan, P. and Wintenberger, O., An invariance principle for weakly dependent stationary general models. Prob. Math. Stat. 27 (2007) 45–73.
Giraitis, L. and Surgailis, D., ARCH-type bilinear models with double long memory. Stoch. Proc. Appl. 100 (2002) 275–300. CrossRef
M.H. Neumann and E. Paparoditis, Goodness-of-fit tests for Markovian time series models. Technical Report No. 16/2005. Department of Mathematics and Statistics, University of Cyprus (2005).
V. Petrov, Limit theorems of probability theory. Clarendon Press, Oxford (1995).
B.L.S. Prakasha Rao, Nonparametric functional estimation. Academic Press, New York (1983).
Rio, E., About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1997) 35–61. CrossRef
E. Rio, Théorie asymptotique pour des processus aléatoires faiblement dépendants. SMAI, Math. Appl. 31 (2000).
Robinson, P.M., Nonparametric estimators for time series. J. Time Ser. Anal. 4 (1983) 185–207. CrossRef
Taqqu, M.S., Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 (1975) 237–302. CrossRef