It is shown that the limiting distribution of any regular estimator of a continuous cdf on [0, 1] can be represented as a convolution of the Brownian bridge process with another distribution on $C\lbrack 0, 1\rbrack$. The result is related to Hajek's representation for limiting distributions of regular parametric estimators.
"Estimating a Distribution Function." Ann. Statist. 5 (2) 400 - 404, March, 1977. https://doi.org/10.1214/aos/1176343806