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