## The Annals of Probability

### Conditions D'Integrabilite Pour Les Multiplicateurs Dans le TLC Banachique

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992447

Digital Object Identifier
doi:10.1214/aop/1176992447

Mathematical Reviews number (MathSciNet)
MR841593

Zentralblatt MATH identifier
0593.60009

JSTOR