Can anyone give me some insight on how to solve this problem? Any help would be greatly appreciated !
Let $(X_n)_{n \geq 1}$ an i.i.d. sequence of real valued r.v.'s with c.d.f. $F$ which is stricly increasing and has two derivatives everywhere. Let $F_n(x) = \frac{1}{n}\sum_{i=1}^n {\bf 1}_{X_i \leq x}$ be the empirical c.d.f., $F_n^{(-1)}(u) = \inf\{x \in \mathbb{R} | F_n(x) \geq u\}$ be the generalized inverse of the empirical cumulative distribution function and $F^{-1}$ be the inverse of $F$. Note that $F_n^{(-1)}(u) = X_{(k_n)}$ where $k_n = \lceil nu \rceil$.
Is this true that under the above hypothesis, \begin{equation} \mathbb{E}\big[\big(F_n^{(-1)}(u) - F^{-1}(u)\big)^2\big] \end{equation} is uniformly bounded?
Thank you