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Let $X_1,\dots,X_n$ be i.i.d with distribution function $F$. 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 $$ \sqrt n\int_{0}^1\left(\left\{\Phi^{-1}(\hat F_n(u))\right\}^2-\left\{\Phi^{-1}(F(u))\right\}^2\right)du\to 0 $$ in probability where $\Phi$ is the commulative standard normal distribution.

It can be proved that $\Phi^{-1}(\hat F_n(x))=\Phi^{-1}(F(x))+O_p(1/\sqrt n)$. But then, I couldn't handle my integral.

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