Annals of Probability

On the Distributions of Sums of Symmetric Random Variables and Vectors

Herman Rubin and Thomas Sellke

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60E99: None of the above, but in this section
Secondary: 62E10: Characterization and structure theory

Keywords
Symmetric random variables sums of random variables two-point distributions symmetric random vectors

Citation

Rubin, Herman; Sellke, Thomas. On the Distributions of Sums of Symmetric Random Variables and Vectors. Ann. Probab. 14 (1986), no. 1, 247--259. doi:10.1214/aop/1176992625. https://projecteuclid.org/euclid.aop/1176992625


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