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July 2002 Necessary and sufficient conditions for the conditional central limit theorem
Jérôme Dedecker, Florence Merlevède
Ann. Probab. 30(3): 1044-1081 (July 2002). DOI: 10.1214/aop/1029867121


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.


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Jérôme Dedecker. Florence Merlevède. "Necessary and sufficient conditions for the conditional central limit theorem." Ann. Probab. 30 (3) 1044 - 1081, July 2002.


Published: July 2002
First available in Project Euclid: 20 August 2002

zbMATH: 1015.60016
MathSciNet: MR1920101
Digital Object Identifier: 10.1214/aop/1029867121

Primary: 60F05
Secondary: 28D05 , 60F17

Keywords: central limit theorem , Invariance principles , stable convergence , Stationary processes , triangular arrays

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 3 • July 2002
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