Prove that for $X_n\sim Beta(n,n)$ $$2\sqrt{2n}\left( X_n-\frac{1}{2}\right)\Rightarrow\mathscr N(0,1)$$ in distribution as $n\rightarrow\infty$.
Let $Y_n$ and $Z_n$ be $Gamma(n,1)$ independent random variables. Then using the facts, that $$X_n=\frac{Y_n}{Y_n+Z_n},$$ and that both $Y_n$ and $Z_n$ are sum of independent $Exp(1)$ random variables, I can apply CLT to those, so both $Y_n$ and $Y_n+Z_n$ are normally distributed. Then I can use the delta method to show, that $X_n$ is normally distributed also.
That is the part where I get confused. Shall I apply the delta method to $Y_n$ with $g(x)=\frac{x}{x+Z_n}$?
Maybe not the method you need/prefer, but if you define $T_n=2\sqrt{2n}(X_n-1/2)$, the pdf of $T_n$ is $$ f_{T_n}(t)=\frac{1}{B(n,n)}\int_0^1 dx\ x^{n-1}(1-x)^{n-1}\delta(t-2\sqrt{2n}x+\sqrt{2n})\ , $$ where $B(\alpha,\beta)$ is a Beta function. Therefore $$ f_{T_n}(t)=\frac{\mathbf{1}_{-\sqrt{2n}<t<\sqrt{2n}}}{2\sqrt{2n}B(n,n)}\left(\frac{t+\sqrt{2n}}{2\sqrt{2n}}\right)^{n-1}\left(1-\frac{t+\sqrt{2n}}{2\sqrt{2n}}\right)^{n-1}\to \frac{e^{-\frac{t^2}{2}}}{\sqrt{2 \pi }}\ , $$ as $n\to\infty$.