Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of $F$ and $F^{-1}_{n}$ is the empirical inverse. For each $N\geq 1$, $q_{N}(t) = \{\sqrt{N}[F^{-1}_{N}(t)-F^{-1}(t)]\}$ is a random variable (r.v). I've already known that, for each $0<t<1$, $$q_n(t)\stackrel{D}{\longrightarrow}X\sim N\left(0,\frac{t(1-t)}{f^2(F^{-1}(t)))}\right).$$ I'm looking for textbooks or papers that approach the (existence of) moments in $q_n(t)$ or its integrability uniform.
Does anybody have suggestions?