The Annals of Probability

Conditions D'Integrabilite Pour Les Multiplicateurs Dans le TLC Banachique

M. Ledoux and M. Talagrand

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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.

Article information

Ann. Probab., Volume 14, Number 3 (1986), 916-921.

First available in Project Euclid: 19 April 2007

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Primary: 60B11: Probability theory on linear topological spaces [See also 28C20]
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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


Ledoux, M.; Talagrand, M. Conditions D'Integrabilite Pour Les Multiplicateurs Dans le TLC Banachique. Ann. Probab. 14 (1986), no. 3, 916--921. doi:10.1214/aop/1176992447.

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