Assume that $X_n\sim Beta(n,n)$. Use the Delta method to show that $$ 2\sqrt{2n}(X_n-\frac{1}{2})\to^d N(0,1) $$
Since $X_n=\frac{Y_n}{Y_n+Z_n}$ where $Y_n$ and $Z_n$ are indepedndent Gamma$(n,1)$ random variables [Why?]. Each of $Y_n$ and $Z_n$ can be written as a sum of exponenetial random variables. From multivariate CLT, we have asymptotic distribution $$ \sqrt{n}([Y_n, \, Z_n]^T-[EY_1, \, EZ_1]^T)=\sqrt{n}([Y_n, \, Z_n]^T-[n, \, n]^T)\to^d N(0, \Sigma) $$ where $\Sigma=\begin{bmatrix} Var[Y_1]&Cov(Z_1, Y_1)\\ Cov(Z_1, Y_1) & Var[Z_1]\end{bmatrix}=\begin{bmatrix} n&0\\ 0 & n\end{bmatrix}$.
From the Delta method, take $f(y,z)=\frac{y}{y+z}$ with $f_y(y,z)=\frac{z}{(y+z)^2}$ and $f_z(y,z)=\frac{-y}{(y+z)^2}$ $$ 2\sqrt{2n}(X_n-\frac{1}{2})=2\sqrt{2n}(f(Y_n, Z_n)-f(n,n))\to^d [\frac{1}{4n}, -\frac{1}{4n}]\times N(0,\Sigma) $$
Question: I got the variance is $1/n$ but not 1?
$$ [\frac{1}{4n}, -\frac{1}{4n}]\begin{bmatrix} n&0\\ 0 & n\end{bmatrix}[\frac{1}{4n}, -\frac{1}{4n}]^T=1/8n $$