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Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Is it true that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $\sum_{i=1}^{n}b_i\alpha_i$ are independent, then $\alpha_1, ..., \alpha_n $ are normal variables?

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    $\begingroup$ Can anyone explain me why was this question closed? $\endgroup$
    – Alexandre Eremenko
    Commented Apr 23, 2017 at 15:48
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    $\begingroup$ It wasn't even a question. $\endgroup$
    – Franz Lemmermeyer
    Commented Apr 23, 2017 at 19:33

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This is called the Darmois-Skitovich theorem. Of course, one needs to add the condition that $a_jb_j\neq 0$.

The reference is MR0346969 Kagan, A. M.; Linnik, Yu. V.; Rao, C. R., Characterization problems in mathematical statistics. Translated from the Russian by B. Ramachandran. John Wiley & Sons, New York-London-Sydney, 1973.

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