Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat F_n(x)=\frac1{n+2}\left(1+\sum_{i=1}^n1_{\{X_i\le x\}}\right) $$ for every $x$, where $1_A$ is the characteristic function of the event $A$.
I wish to prove $$ \sqrt n\int_{0}^1\Phi^{-1}(u)\left(\Phi^{-1}(\hat F_n(u))-\Phi^{-1}(u)\right)du\to 0 $$ in probability as $n\to\infty$ with $\Phi$ is the cumulative normal distribution function.
It can be proved that $\Phi^{-1}(\hat F_n(x))=\Phi^{-1}(u)+O_p(1/\sqrt n)$. But then, I couldn't handle my integral.