How to prove that $\sqrt{n}(\ln S_n-\mu)$ converges in law to $\mathcal N(0,\sigma^2)$ where$S_n$is the logarithm of geometric mean of some iid $X_i$?

Question

Let $X_n$ be some positive random variables, independent and identically distributed with $(\ln{X_n})^2$ integrable, $\textbf{E}[\ln X_n]=\mu$ and $\textbf{Var}[\ln{X_n}]=\sigma^2$. Define the geometric mean $S_n=(X_1X_2\dots X_n)^{\frac{1}{n}}$. We want to prove that $\sqrt{n}(\ln S_n-\mu)$ converges in law to $\mathcal N(0,\sigma^2)$. My guess is that this is an application of $Lindeberg$'s CLT. However, I wasn't able to prove the condition.

Answer 1

$\sqrt{n}(\ln S_n-\mu)=\sqrt{n}(\frac{1}{n}\sum\limits_{i=1}^{n}\ln X_i-\mu)$ and $\ln X_i$ i.i.d. r.v.s, and the result follows from CLT, i.e. it converges in distribution to $\mathcal{N}(0,\sigma^2)$, since Var[$\ln X_i]=\sigma^2$.