Let $\{X_n\}_{n=1}^{+\infty}$ be a sequence of i.i.d. random variables with expectation $\mathbb{E}{X_1} = \mu$ and variance $\text{Var}({X_1}) = \sigma^2$. Now consider a sequence $S_n = X_1 + X_2 + \dots + X_n$. Where does $\sqrt{n}\left(\frac{n}{S_n} - \frac{1}{\mu}\right)$ approach by distribution?
I think I should use the central limit theorem (CLT) here, but I'm not sure how can I use the fact that $\dfrac{S_n - n\mu}{\sigma\sqrt{n}} \to Z \sim \mathcal{N}(0, 1)$ by distribution.
Any ideas or help on how to approach this problem would be greatly appreciated.
From CLT, you have
$$\sqrt{n}\left( \frac{S_n}{n} - \mu\right) \xrightarrow{n \to+\infty} N(0,\sigma^2)$$
Apply the delta method with $g(x) = \frac{1}{x}$, you get
$$\sqrt{n}\left( \frac{n}{S_n} - \frac{1}{\mu}\right) \xrightarrow{n \to+\infty} N(0,\sigma^2 \frac{1}{\mu^4})$$