We prove a Donsker-type invariance principle for a nearest-neighbor-type conditional empirical process. As an application we show asymptotic normality of conditional quantiles and derive large-sample distribution-free tests and confidence bands for a conditional distribution function.
"Conditional Empirical Processes." Ann. Statist. 14 (2) 638 - 647, June, 1986. https://doi.org/10.1214/aos/1176349943