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?



A problem about normal distribution, independent random variables
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. 


