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 $X$.

You could try with cumulants, which are additive with respecto to (independent) addition.
Say $Y = X_1 + X_2$ where $X_i$ are iid.
Thus, if you are given the distribution of X, you compute its cumulants, $\kappa^{[x]}_n$, and you have $\kappa^{[y]}_n = 2 \kappa^{[x]}_n$ . For this you can (formally, not necesarily easily)  recover the distribution of $Y$.
Of course, this only can be applied when $X$ has finite moments. And even then, it does not guarantee that the new cumulants are "valid" (i.e. they correspond to a valid probability function). I guess this could be worked, though.
