There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail:
Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.
In particular, how come we are guaranteed $X^\sim$ and $Y^\sim$ such that $X^\sim \sim X$ and $Y^\sim \sim Y$ with $X^\sim$ and $Y^\sim$ independent?