X=Y+Z, know the distribution of X, under which conditions can I recover(pin down) the distribution of Y and Z?

Question

You may add any conditions you want if this is doable. Otherwise, it would be great if you could provide any impossibility result.

Also greatly appreciate it if you could guide me to related problems, results or papers. Many thanks:)

Answer 1

In general you can't say which distributions gave you $X$.

However, with some additional restrictions you can tell something about the family of that distribution.

An example is family of Gaussian distributions. Famous Cramer's theorem says that if $X$ has normal distribution, then $Y$ and $Z$ also must have normal distribution.

See Cramer's theorem

Also there is Raikov theorem which tells the same for Poisson distributed random variable. And there is theorem about sums of normal and Poisson distribution. So that if you know that $X$ has the same distribution as Poisson+Normal, then $Y$ and $Z$ are necessarily like that (one of them is Poisson one is normal).

Hope that helped.