Show that $\frac{2}{\sigma}(\sqrt{S_n}-\sqrt{n}) \rightarrow Z$ in distribution with $Z \sim N(0,1)$

Question

Let $X_n$ be a sequence of non negative random variables independent and identically distribuited such that $E[X_1]=1$ and $Var(X_1)=\sigma ^2 < \infty$ define $S_n=\sum_{k=1}^{n} X_k$ show that $$\frac{2}{\sigma}(\sqrt{S_n}-\sqrt{n}) \rightarrow Z$$ in distribution with $Z \sim N(0,1)$

I try to use characteristic functions and the central limit theorem but i not sure to these is the right way, one hint is rewrite as $$ \frac{S_n -n}{\sqrt{n}}=\frac{(\sqrt{S_n}+\sqrt{n})(\sqrt{S_n}-\sqrt{n})}{\sqrt{n}}$$ any hint or help i will be very grateful

Answer 1

Hint: $\frac {\sqrt {S_n} +\sqrt n} {\sqrt n} =\sqrt {S_n/n} +1 \to 2$ almost surely by SLLN.

Answer 2

Let $\bar{X} = S_n/n$. The CLT implies that $\sqrt n(\bar{X}-1)\overset{D}{\to} N(0,\sigma^2)$. Then use the delta method to deduce that $$ \sqrt S_n-\sqrt n = \sqrt n(\sqrt{\bar{X}}-1)\overset{D}{\to} N(0,\sigma^2\times2^{-2}) $$ Hence $$ 2\times\frac{\sqrt S_n-\sqrt n}{\sigma} \overset{D}{\to} N(0,1) $$ by Slutsky's theorem for example.