Dependent Lindeberg central limit theorem and some applications

Jean-Marc Bardet (1, 2) , Paul Doukhan (1, 2, 3) , Gabriel Lang (4) , Nicolas Ragache (3)

1 SAMOS - Statistique Appliquée et MOdélisation Stochastique
2 CES - Centre d'économie de la Sorbonne
3 CREST - Centre de Recherche en Économie et Statistique
4 MIA-Paris - Mathématiques et Informatique Appliquées

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.

Mots clés: Weak dependence; Central limit theorem; Lindeberg method; Kernel density estimation; Subsampling

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jean-Marc Bardet, Paul Doukhan, Gabriel Lang, Nicolas Ragache. Dependent Lindeberg central limit theorem and some applications. ESAIM: Probability and Statistics, 2008, 12, pp.154-172. ⟨10.1051/ps:2007053⟩. ⟨hal-00127903⟩