We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\sigma^2)$ for some $\mu$ and $\sigma$, can we conclude that both $X$ and $Y$ obey normal distribution? or under some conditions added on $p\_X$ and $p\_Y$ (the density functions of $X$ and $Y$)?



Sum of independent random variables
Yes, and the same holds for Poisson, and for mixtures of Gauss and Poisson.
All these are special cases of the general question: if $X\_j$ are independent
and we know the distribution of their sum, what can be said about the distributions of the $X\_j$. This general question is addressed in the book
Linnik, Ostrovskii, Decomposition of random variables and vectors, AMS 1977
(translation from the Russian).


