Suppose we have $X_1,\cdots,X_n$ be a random sample with $X_i\backsim \chi^2(1)$. I'd like to show that as $n\rightarrow \infty$, $\frac{\bar{X}-1}{\sqrt{\frac{2}{n}}}$ is standard normal.
Here's my attempt.
Since $X_i\backsim \chi^2(1)$, we know that $\mathbb{E}(X_i)=1, \text{Var}(X_i)=2.$ As $n\rightarrow \infty $,the Central Limit Theorem says that, $\bar{X}$ is approximately $N(1,\frac{2}{n})$. So $ \frac{\bar{X}-1}{\sqrt{\frac{2}{n}}}\rightarrow Z\backsim N(0,1).$
Does my attempt pass as a genuine solution? Is there way of doing this using Mgf's?
Thanks.