Central Limit Theorem and Strong Law of Large Numbers. Proof that converges in distribution to $N(0, e^2)$ I have this exercise of my homework.
Let $\{ X_n: n \geq 1\}$ be independent random variables and identically distribuited with uniform distribution $U(0,1)$ and $Y_n=(\prod_{i=1}^n X_i)^{-1/n} $.
Show that $\sqrt{n} (Y_n-e)$ converges in distribution to $N(0, e^2)$
I only know that $Y_n$ converges almost surely to $e$.
I do not know to do the proof using central limit theorem.
I know the law of large numbers.
I think that you can use delta method.

One direction that sounds promising is to note that
$$
\ln Y_n
 = \ln \left( \left(\prod_{k=1}^n X_k\right)^{-1/n}\right)
 = \frac{-1}{n} \sum_{k=1}^n \ln X_k
 = \frac{1}{n} \sum_{k=1}^n L_k,
$$
where $L_k = -\ln X_k$ and now you can apply the CLT directly.
