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January, 1986 On the Distributions of Sums of Symmetric Random Variables and Vectors
Herman Rubin, Thomas Sellke
Ann. Probab. 14(1): 247-259 (January, 1986). DOI: 10.1214/aop/1176992625


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.


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Herman Rubin. Thomas Sellke. "On the Distributions of Sums of Symmetric Random Variables and Vectors." Ann. Probab. 14 (1) 247 - 259, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0624.60025
MathSciNet: MR815968
Digital Object Identifier: 10.1214/aop/1176992625

Primary: 60E99
Secondary: 62E10

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

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 1 • January, 1986
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