I found an old exam in stochastic Analysis but without a solution. Here is the exercise: Let $X_{1},X_{2},...$ be i.i.d. with $\mathbb{P}(X_{n} = k) = 2^{-k}$ for $k \in \mathbb{N}$. Define $Z_{0} = 1$ and \begin{equation} Z_{n} = 3Z_{n-1}/2^{X_{n}},~~n\geq1. \end{equation}
(a) Show that $(Z_{n})_{n\geq0}$ is a martingal with respect to the filtration $\sigma(X_{1},...,X_{n}))_{n\geq0}$
(b) Show $Z_{n} \rightarrow 0$ almost surely
I have done part (a). But how can I Show part (b)?
By iteration we get $Z_n=\frac {3^{n}} {2^{S_n}}$ for $n \geq 2%+$ where $S_n=X_2+X_3+...+X_n$. By Strong Law $Z_n^{1/n} \to \frac 3 {2^{EX_1}}=3/4$ almost surely. This implies that $Z_n \to 0$ a.s..