Under the condition that the minimal sufficient statistics are transitive, the sequence of Rao-Blackwell estimators of distribution function has been shown to form a reverse martingale sequence. Weak convergence of the corresponding empirical process to a Gaussian process has been established by assuming that the sufficient statistics are $U$-statistics and utilizing certain results on the convergence of conditional expectations of functions of $U$-statistics along with the functional central limit theorems for (reverse) martingales by Loynes (1970) and Brown (1971).
@article{1176995813,
author = {Bhattacharyya, B. B. and Sen, P. K.},
title = {Weak Convergence of the Rao-Blackwell Estimator of a Distribution Function},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 500-510},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995813}
}
Bhattacharyya, B. B.; Sen, P. K. Weak Convergence of the Rao-Blackwell Estimator of a Distribution Function. Ann. Probab., Tome 5 (1977) no. 6, pp. 500-510. http://gdmltest.u-ga.fr/item/1176995813/