Sufficient conditions are developed for various versions of the weighted empirical quantile process to converge weakly in $L^2(0, 1)$ to a weighted Brownian bridge. The results are directly applicable to the derivation of the asymptotic distribution of goodness of fit tests based on the sample quantiles that can be written as a functional defined on $L^2(0, 1)$ continuous in the norm topology. In the process, tight bounds for the moments of transformed uniform order statistics are derived that are likely to have applications elsewhere.
"Weak Convergence of the Weighted Empirical Quantile Process in $L^2(0, 1)$." Ann. Probab. 12 (1) 243 - 255, February, 1984. https://doi.org/10.1214/aop/1176993387