Application of Central Limit Theorem for nonnegative RV

Question

Suppse $X_1, X2, \dots$ are iid nonnegative r.v.s with mean 1 and finite variance $\sigma^2>0$. Show that $2(\sqrt{S_n}-\sqrt{n}) \rightarrow \mathcal{N}(0,\sigma^2)$

Answer 1

Hint: $$\sqrt{S_n}-\sqrt n=\frac{S_n-n}{\sqrt{S_n}+\sqrt n}=\color{blue}{\frac{\sum_{j=1}^n(X_j-\mathbb E(X_j))}{\sqrt n\sigma}}\color{red}{\frac{\sigma\sqrt n}{\sqrt{S_n}+\sqrt n}}.$$ The convergence of the blue term follows from the classical central limit theorem while the red term can be treated with the law of large numbers.

To combine the two results, we will need Slutsky's lemma.

Answer 2

Apply delta method:

$$2(\sqrt{S_n} - \sqrt{n}) =2\sqrt{n} \left[\sqrt{\frac{S_n}n} - 1 \right]\\ =2\sqrt{n} \left[f\left(\frac{S_n}n\right) - f(1) \right] $$with $f(x) = \sqrt{x}$ $$ \to 2 f'(1)N(0,\sigma^2)=N(0,\sigma^2) $$

in distribution.