Consider $X_1,X_2,\dots,X_n,\dots$ as a sequence of independent random variables with probability generating functions $F_1(s),F_2(s),\dots,F_n(s),\dots$. It is well known that for $n\in \mathbb{N}$ the probability generating functions of $\sum_{i=1}^{n}X_i$ is $\prod_{i=1}^{n}F_i(s)$.
Can somebody help me how it works, when we consider $\sum_{i=1}^{\infty}X_i$? Can I under some condition say that the probability generating functions of $\sum_{i=1}^{\infty}X_i$ is $\prod_{i=1}^{\infty}F_i(s)$.
Any help will be appreciated.
A series of independent r.v 's converges almost surely iff it converges in probability iff it converges in distribution. If $\sum X_i$ convegence in any of these senses then the generating function of $\sum X_i$ is the product of the generating functions of the $X_i$'s (by DCT).