The Annals of Probability

Necessary and sufficient conditions for the conditional central limit theorem

Jérôme Dedecker and Florence Merlevède

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Abstract

Following Lindeberg's approach, we obtain a new condition for a stationary sequence of square-integrable and real-valued random variables to satisfy the central limit theorem. In the adapted case, this condition is weaker than any projective criterion derived from Gordin's theorem [\textit{Dokl. Akad. Nauk SSSR} \textbf{188} (1969) 739--741] about approximating martingales. Moreover, our criterion is equivalent to the {\it conditional central limit theorem}, which implies stable convergence (in the sense of Rényi) to a mixture of normal distributions. We also establish functional and triangular versions of this theorem. From these general results, we derive sufficient conditions which are easier to verify and may be compared to other results in the literature. To be complete, we present an application to kernel density estimators for some classes of discrete ime processes.

Article information

Source
Ann. Probab., Volume 30, Number 3 (2002), 1044-1081.

Dates
First available in Project Euclid: 20 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1029867121

Digital Object Identifier
doi:10.1214/aop/1029867121

Mathematical Reviews number (MathSciNet)
MR1920101

Zentralblatt MATH identifier
1015.60016

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 28D05: Measure-preserving transformations

Keywords
Central limit theorem stable convergence invariance principles stationary processes triangular arrays

Citation

Dedecker, Jérôme; Merlevède, Florence. Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002), no. 3, 1044--1081. doi:10.1214/aop/1029867121. https://projecteuclid.org/euclid.aop/1029867121


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