Suppose that $X_1, \cdots, X_n$ is a Uniform$[0,1]$ data sample, for large $n$.
By the Central Limit Theorem we know that $\sqrt{n}(\bar{X} - 1/2) \rightarrow^D N(0, 1/12)$.
If $E[X^4] \lt \infty$, we can show that $\sqrt{n}(S^2 - 1/12) \rightarrow^D N(0, Var[(X_1 - 1/2)^2])$, where $S^2 = (n -1)^{-1}\sum^n_1 (X_i - \bar{X})^2$.
By the Delta Method we know that $\sqrt{n}(S - 1/\sqrt{12}) \rightarrow^D N(0, \frac{Var[(X_1 - 1/2)^2]}{48}).$
I have two concerns:
$\textbf{1.}$ I want to find the large-sample limiting joint distribution of $\sqrt{n}(\bar{X} - 1/2)$ and $\sqrt{n}(S^2 - 1/12).$ I know that $\bar{X}$ and $S^{2}$ are independent for normal distributions. Are they still independent for a Uniform$[0, 1]$ distribution?
$\textbf{2.}$ I also want to find the large-sample limiting distribution of $\sqrt{n}(\bar{X}/S - \sqrt{3})$. I have tried to find a way to use the Delta Method but had no success.
Any hints will be appreciated.
