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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 $X$.

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    Risking at stating triviality: if X is infinitely divisible distribution, then it does admit such a decomposition, but the class you seek to characterize is definitely wider. Skimming though Lukacs' book, he says that if $\phi(t)$ is analytic in a strip $-\alpha < \operatorname{Im}(t) < \beta$ and does not vanish within it, it can be decomposed into a product of two c.f. also analytic in that strip, as far as I understood.2011-08-19

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