We consider Bahadur-Kiefer representations for smoothed quantile processes. We prove that the asymptotics of the distance between smoothed empirical and quantile processes can be completely different from that of the unsmoothed ones. We obtain a complete characterization of the possible limits.
"On the Distance Between Smoothed Empirical and Quantile Processes." Ann. Statist. 23 (1) 113 - 131, February, 1995. https://doi.org/10.1214/aos/1176324458