Abstract
There exist independent random variables $X_1$ and $X_2$ such that $X_1$ is symmetric, $X_2$ is not symmetric, but $X_1 + X_2$ is symmetric. If $X_1$ and $X_2$ are i.i.d. random variables with a fractional moment and if for all real $\alpha P\lbrack X_1 + \alpha X_2 > 0\rbrack = \frac{1}{2}$ then they are symmetric.
Citation
David L. Burdick. "A Note on Symmetric Random Variables." Ann. Math. Statist. 43 (6) 2039 - 2040, December, 1972. https://doi.org/10.1214/aoms/1177690880
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