If $X$ poisson and $X=X_1+X_2$, then integer valued $X_1$ and $X_2$ are also poisson.

Question

I know if $X$ and $Y$ are poisson, then $X + Y$ is also poisson. But now I met a strange problem which states that if $X$ is poisson, $X_1 + X_2 = X$ and $X_1$,$X_2$ are independent nonnegative integer valued then both $X_1$ and $X_2$ are also poisson. I try to use the characteristic function to solve this problem but didn't get any helpful result.

Answer 1

This is known as Raikov's theorem. A proof can be found here (Theorem 8.2.2).