## Annals of Probability

### On the Distributions of Sums of Symmetric Random Variables and Vectors

#### Abstract

Let $F$ be a probability distribution on $\mathbb{R}$. Then there exist symmetric (about zero) random variables $X$ and $Y$ whose sum has distribution $F$ if and only if $F$ has mean zero or no mean (finite or infinite). Now suppose $F$ is a probability distribution on $\mathbb{R}^n$. There exist spherically symmetric (about the origin) random vectors $\mathbf{X}$ and $\mathbf{Y}$ whose sum $\mathbf{X + Y}$ has distribution $F$ if and only if all the one-dimensional distributions obtained by projecting $F$ onto lines through the origin have either mean zero or no mean.

#### Article information

Source
Ann. Probab., Volume 14, Number 1 (1986), 247-259.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992625

Digital Object Identifier
doi:10.1214/aop/1176992625

Mathematical Reviews number (MathSciNet)
MR815968

Zentralblatt MATH identifier
0624.60025

JSTOR