Inference, including variance estimation, can be made from stratified samples by selecting a balanced set of subsamples. This balanced subsampling method is generically called the balanced repeated replication method in survey data analysis, which includes McCarthy's balanced half-samples method and its extensions for more general stratified designs. We establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile. The consistency results also hold when balanced subsampling is replaced by random subsampling. As a key technical prerequisite, we prove a Bahadur-type representation for sample quantiles in stratified random sampling.
"Asymptotic Properties of the Balanced Repeated Replication Method for Sample Quantiles." Ann. Statist. 20 (3) 1571 - 1593, September, 1992. https://doi.org/10.1214/aos/1176348785