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Issue ESAIM: PS
Volume 12
Page(s) 154 - 172
DOI 10.1051/ps:2007053
Published online 23 January 2008

ESAIM: PS, 2008, Vol. 12, p. 154-172
DOI: 10.1051/ps:2007053

Dependent Lindeberg central limit theorem and some applications

Jean-Marc Bardet1, Paul Doukhan1, 2, Gabriel Lang3 and Nicolas Ragache2

1  Samos-Matisse-CES, Université Panthéon-Sorbonne, 90 rue de Tolbiac, 75013 Paris, France.
2  LS-CREST, Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, France.
3  AgroParisTech, UMR MIA 518 (AgroParisTech-INRA), 75005 Paris, France.


Received June 15, 2007. Revised June 27, 2007. Published online 23 January 2008

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($\infty$), bilinear, Volterra processes, $\ldots$, enter this frame.


Mathematics Subject Classification. 60F05, 62G07, 62M10, 62G09

Key words: Central limit theorem, Lindeberg method, weak dependence, kernel density estimation, subsampling


© EDP Sciences, SMAI 2008


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