Let $ (X)_{n\in\mathbb{N}}$ be i.i.d random variables with $P(X_j=1/2)=P(X_j=3/2)=1/2$. Also $Z_n:= \prod_{i=1}^n X_i$. Determine $\lim \limits_{n \to \infty}Z_n $ a.s.
I'm relatively new to this so I'm not quite sure how to approach this one, can anyone tell me how to solve this or give me a hint in the right direction?
As commented by @Jochen, the easiest way is to take logarithm.
Let $$Y_n=\log Z_n = \sum_{i=1}^{n}\log X_i$$ Then $$\mathbb{E}[\log X_i] = \frac12\log\frac12+\frac12\log\frac32=\frac12\log\frac34<0$$ Using the SLLN, $$\frac{Y_n}{n}=\frac1n\sum_{i=1}^{n}\log X_i\xrightarrow{a.s.}\mathbb{E}[\log X_i]<0$$ $$Z_n = e^{n\cdot\frac{Y_n}{n}}\xrightarrow{a.s.}0$$