TITLE: Product of Uniform Distribution
QUESTION [1 upvotes]: I know that there exists some discussions related to my question, however, I couldn't find an explanation for my question. I hope it is not a duplicate.

Let $X_n$ be sequence of i.i.d. uniform distributions on $(0,a)$, and define $Y_n = \prod^n_k X_k$. 
Problem
  For what values of $a$, $\lim Y_n\to 0$ a.s.

Attempt
Note that,
\begin{equation}
Y_n = \exp\left(n\times\frac{1}{n}\sum_k^n \log(X_k)\right)
\end{equation}
and by SLLN, if $E\log(X_1)<0$ it follows that $Y_n\to 0$ a.s., which is true if $a<e$. 
Question How can I discuss $a=e$?

REPLY [0 votes]: Copied from the comment of Did,
If $a=e$, then $S_n = \sum_{k=1}^n \log X_k$ defines a random walk on the real line with centered integrable steps, hence $(S_n)$ is recurrent, which implies that $(Y_n)$ is almost surely unbounded. In particular, $P(Y_n \to 0) = 0$.
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