I am given that $X_i$ are iid Unif(0,1), $i=1,\cdots,n$. $Y_n$ is defined as $Y_n=\left(\prod_{i=1}^n X_i\right)^{-1/n}$. I am to show that $\sqrt n (Y_n-e)$ converges in distribution to an $N(0,e^2)$ random variable as $n\to\infty$.
So far, I've mostly been trying to use log to transform this into something that looks like the central limit theorem will apply. However, I don't seem to make any headway that way. I can show, e.g., that $$\mathbb P(\sqrt n (Y_n-e)<x) = \mathbb P\left(\sum_{i=1}^n \log X_i > \log\left( \left(\frac x {\sqrt n} + e\right)^{-n} \right )\right)$$ But I can't see what to do with this. Am I on the right track? If so, how should I proceed?
Use CLT to show that $\sqrt{n} \left( Z_n -1 \right) \xrightarrow{d} N(0,1)$, where $$ Z_n = \log (Y_n) = \frac{1}{n} \sum_{i=1}^n - \log(X_i) $$ then, use delta-method (https://en.wikipedia.org/wiki/Delta_method) with $g : x \mapsto e^x$ to show that $\sqrt{n}(Y_n - e) \xrightarrow{d} N(0,e^2)$.