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July, 1986 Conditions D'Integrabilite Pour Les Multiplicateurs Dans le TLC Banachique
M. Ledoux, M. Talagrand
Ann. Probab. 14(3): 916-921 (July, 1986). DOI: 10.1214/aop/1176992447


Let $X$ be a Banach space valued random variable satisfying the central limit theorem and $\xi$ be a real valued random variable, independent of $X$. If $\xi$ is in the Lorentz space $L_{2,1}$, the product $\xi X$ satisfies the central limit theorem. We show that this condition on $\xi$ cannot be improved, characterizing $L_{2,1}$ as the space of all random variables $\xi$ such that the preceding implication holds for all vector valued $X$ satisfying the central limit theorem. In particular, there exist independent random variables $X$ and $\xi$ both satisfying the central limit theorem such that $\xi X$ does not.


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M. Ledoux. M. Talagrand. "Conditions D'Integrabilite Pour Les Multiplicateurs Dans le TLC Banachique." Ann. Probab. 14 (3) 916 - 921, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60009
MathSciNet: MR841593
Digital Object Identifier: 10.1214/aop/1176992447

Primary: 60B11
Secondary: 46E30 , 60B12

Keywords: espace de Lorentz $L_{2,1}$ , multiplicateurs , Theoreme limite central

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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