Does this converge?

Question

If I have $$X_i=\begin{cases}2\quad p=\frac{1}{3}\\ \frac{1}{2}\quad p=\frac{2}{3} \end{cases}$$ random variables with the same distribution. How can I compute the limit almost sure as $n\to\infty$ of $\prod_{i=1}^nX_i$? I just need a hint. Thanks.

Answer 1

In this solution, I assume that $P\left(X_i=2\right)=\frac{1}{3}$. In this case, define $Y_i=\log_2 X_i$; then $$Y_i=\left\{\begin{matrix}1,&p=\frac{1}{3}\\-1,&p=\frac{2}{3}\end{matrix}\right.$$ Obviously, $$\prod_{i=1}^n X_i=2^{\sum_{i=1}^n Y_i}$$ so it is enough to check the sum of $Y_i$. Now, can you prove what their sum is a.s.?