Let $\xi_p$ be pth quantile of the distribution $F(x)$ with derivative at $\xi_p$, $f(\xi_p) >0$. Then,
$$ |\hat\xi_{p,n} - \xi_p| \leq \frac{2}{f(\xi_p)}\sqrt{\frac{\log n}{n}} $$
almost surely for large enough $n$ and any $p \in (0,1)$, where $\hat\xi_{p,n}$ is the pth sample quantile from an empirical distribution $F_n(x)$ for $F(x)$ with $1/n$ mass for each observation.
I have tried Taylor expansion to get a bound but this approach does not seem to work; it seems like I need to rather work with empirical distribution (using Hoeffding's inequality) rather than sample quantile to derive the term $\log n$, but I am not sure of how I can derive this upper bound.
