# mgf of an infinite sum of independent random variables

It is known that if $S_n = \sum_{i=1}^{n} X_i$, where the $X_i$ are independent random variables, then the moment-generating function for $S_n$ is given by $M_{S_n}(t)=\prod_{i=1}^{n}M_{X_i}(t)$.

Now suppose that we have infinitely many random variables $X_i,i\in\mathbb{N}$, and suppose that the moment-generating function exists for each random variable $X_i,i\in\mathbb{N}$. Denote $S = \sum_{i=1}^{\infty} X_i$. Then do we have that (maybe under some additional conditions) $M_{S}(t)=\prod_{i=1}^{\infty}M_{X_i}(t)$? If so then how can we prove it? I really tried hard to find out such a formula for the mgf of an infinite sum of independent random variables but all the lecture notes and books that I dipped into only present the formula for finite case.

May I suggest you try to do this in the case where $X_i = \pm 2^{-i},$ with random signs, as an exercise?