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.
Publié le : 1986-01-14
Classification:
Symmetric random variables,
sums of random variables,
two-point distributions,
symmetric random vectors,
60E99,
62E10
@article{1176992625,
author = {Rubin, Herman and Sellke, Thomas},
title = {On the Distributions of Sums of Symmetric Random Variables and Vectors},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 247-259},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992625}
}
Rubin, Herman; Sellke, Thomas. On the Distributions of Sums of Symmetric Random Variables and Vectors. Ann. Probab., Tome 14 (1986) no. 4, pp. 247-259. http://gdmltest.u-ga.fr/item/1176992625/