Dependent Lindeberg central limit theorem and some applications

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: 15 June 2007
Revised: 27 June 2007

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.