TITLE: A question on Cramer's theorem
QUESTION [9 upvotes]: Almost everybody is familiar with Cramer's theorem: a sum $X+Y$ of of independent random variables is normal if and only if both $X$ and $Y$ are normal. Are there any other classes of distributions which can replace normality in the above statement?

REPLY [8 votes]: As has been already stated, the analogous statement (I believe) is known to hold for stable distributions (I can't find a precise reference for this, but results along these lines are discussed in sections 6 and 7 of this survey article of E. Lukacs). 
It is also worth mentioning that there is an longstanding problem of Mark Kac which seeks to extend the spirit of these results to a more general setting. Note that we may reformulate Cramer's theorem as follows: 
Theorem (Cramer)
Let $\sigma_1$ and $\sigma_2$ be positive functions such that: 
$$\int_{-\infty}^{\infty} e(-\xi x) d\sigma_1 (x) \times \int_{-\infty}^{\infty} e(-\xi x) d\sigma_2(x) = e^{-\xi^2/4}.$$
Then $\sigma_1(x)$ and $\sigma_2$ are Gaussians.
Seeking to generalize this, Kac's problem is the following:


(Kac's Problem) Let $\phi(x,y) = \frac{1}{ \frac{1}{x} +\frac{1}{y} -1} $ and assume that 
    $$ \phi \left( \int_{-\infty}^{\infty} e(-\xi x) d\sigma_1 (x), \int_{-\infty}^{\infty} e(-\xi x) d \sigma_2(x)\right) = \frac{1}{1+\xi^2}.$$ Is it true that $\sigma_1(x) = \alpha_1 e^{-\beta_1 |x|}$ and  $\sigma_2(x) = \alpha_2 e^{-\beta_2 |x|}$?


This is problem 178 from the Scottish book, where it is dated September 11, 1938.