Suppose $\hat{F}_{n}$ is the empirical distribution function based on a sample $(X_{1},\ldots,X_{n})$, where each $X_{i}$ has distribution function $F$.
Also, suppose that the distribution of $X_{i}$ has a smooth density $f(x)$. Let $\xi_{p}$ be the $p^{th}$ quantile of $X_{i}$, that is, $F(\xi_{p}) = p$.
Is there anything known about the following process: $$ \sup_{1 \leq i \leq n} \frac{\hat{F}_{n}(\xi_{i/n}) - F(\xi_{i/n})}{f(\xi_{i/n})} $$ For example, does the above supremum go to zero in probablity at a certain rate?
I know that from the Kiefer-Wolfowitz inequality that
$$ P(\sup_{x}|\hat{F}_{n}(x) - F(x)| \geq \varepsilon) \leq 2e^{-2n\varepsilon^{2}}, $$ but I don't know what happens when we divide by the density.