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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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