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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: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan and Louhichi (1999), a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal or non causal time 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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