The set of independent and identically distributed r.v. $Y_{1},Y_{2},...,Y_{n}$ has a distribution $Pr(Y_i=1)=Pr(Y_i=1/2)=1/2\ \forall i>0$ . If $Z_n=\prod_{i=1}^{n}Y_i$ and $S_n=\sum_{i=1}^{n}Z_i \ \forall n>0$. Does $Sn$ converge a.s. to some limit r.v.? If so, find the mean and variance of the limit r.v.
I used the properties of iid r.v. to rewrite Sn as following: $$S_n=\sum_{i=1}^{n}\prod_{i=1}^{n}Y_i=\sum_{i=1}^{n}(Y_1)^{n}=(Y_1)+(Y_1)^{2}+...+(Y_1)^{n}.$$
According to the distribution, Yi=1 or Yi=1/2 but I am not sure how to implement that fact here. Moreover, the Law of Large Numbers and a.s. convergence involve the sum of r.v., so I don't know if they can apply here as well.
Note that $S_n - S_{n-1} = Z_n$. Use this to show that $S_n$ is a Cauchy sequence in $L^2$, thus $S_n$ has an $L^2$ limit $S_{\infty}$. Then $S_n$ must converge to $S_{\infty}$ almost surely along some subsequence. But since $S_n$ is a monotonic sequence, convergence along a subsequence implies convergence for the entire sequence.