The weak convergence of empirical distribution functions subject to random perturbations and scale factors to a Gaussian process is established. This result is used to study the efficiencies of tests based on spacings in goodness-of-fit problems.
"Weak Convergence of Empirical Distribution Functions of Random Variables Subject to Perturbations and Scale Factors." Ann. Statist. 3 (2) 299 - 313, March, 1975. https://doi.org/10.1214/aos/1176343058