Let $U_1, U_2,\ldots,U_n$ be iid uniform random variables on $[0,1]$.
$U_{1,n}\leq U_{2,n}\leq\cdots\leq U_{n,n}$ be the order statistics.
Show that, as $\frac{k_n}{n}\to p$ and $0\leq p\leq 1$ $$\frac{\sqrt{n}(U_{k_n,n}-\frac{k_n}{n+1})}{p(1-p)^{1/2}} \to N(0,1)$$
This is what I have tried:
Using Renyi Rerepresentation:
$$U_{k_n,n} = \frac{S(k_n)}{S(n+1)}$$
$$U_{k_n,n}-\frac{k_n}{n+1}=\frac{S(k_n)-k_n}{S(n+1)}-\frac{(S(n+1)-(n+1))k_n}{(n+1)S(n+1)}$$
This is where I get stuck
Eventually I want to get to something like $\frac{S(k_n)-k_n}{\sqrt{k_n}}$ multiply by other fractions, those fractions converge to 1 and the whole transformation converges to $N(0,1)$