The problem I am working on is the following. Let $X_1, X_2; ...$ be i.i.d. random variables with uniform distribution on $[0; 1]$. We want to show that $\sum_{n=1}^{\infty}\prod_{i=1}^{n} X_i$ is finite almost surely, i.e. that $$P\left( \sum_{n=1}^{\infty}\prod_{i=1}^{n} X_i < \infty\right) = 1$$.
What I have been doing is defining $Y_n = \prod_{i=1}^{n} X_i$ and now I am trying to prove that $Y_n$ is distributed as a Gamma (a result I found in some textbook). At this point, I aim at showing the the sum of Gamma random variables is a Gamma random variable as well. Finally, I should show that a Gamma variable is finite almost surely, but I am not sure on how to procede on this.
Plus, is there, in your opinion, a more efficient way of proceeding?
Thank you in advance
Since the terms in the sum are non-negative almost surely,
$$E(\sum_{n=1}^{\infty}\prod_{i=1}^{n} X_i) = \sum_{n=1}^{\infty}E(\prod_{i=1}^{n} X_i) = \sum_{n=1}^{\infty}\prod_{i=1}^{n} E(X_i)= \sum_{n=1}^{\infty}\frac 1{2^n} <\infty $$
Since $\sum_{n=1}^{\infty}\prod_{i=1}^{n} X_i$ is non-negative almost surely and has finite expectation, it is finite almost surely.
All this can be rewritten in the framework of measure theory using integrals instead of $E$. You might be more familiar with the results I used in a measure-theoretic outlook.