When can a random variable be written as a sum of two iid random variables? Suppose $X$ is a random variable; when do there exist two random variables $X',X''$, independent and identically distributed, such that $$X = X' + X''$$ 
My natural impulse here is to use Bochner's theorem but that did not seem to lead anywhere. Specifically, the characteristic function of $X$, which I will call $\phi(t)$, must have the property that we can a find a square root of it (e.g., some $\psi(t)$ with $\psi^2=\phi$) which is positive definite. This is as far as I got, and its pretty unenlightening - I am not sure when this can and can't be done. I am hoping there is a better answer that allows one to answer this question merely by glancing at the distribution of $X$.

This Wikipedia article addresses precisely this question, with some nice examples, but doesn't actually answer the question: http://en.wikipedia.org/wiki/Indecomposable_distribution
There is a book titled Characteristic Functions, by Eugene Lukacs, which addressed many questions of this kind.
If you want to address only infinite divisibility, rather than the (possibly more complicated) questions of finite divisibility, then one necessary condition for infinite divisibility is that all of the even-order cumulants must be non-negative. PS: That all of the even-order cumulants must be non-negative is true of compound Poisson distributions, but now I think probably not of infinitely divisible distributions in general.
