In this paper two convergence theorems are proved. One gives a strong law of large numbers for a class of linear rank statistics and the other gives weak convergence of a weighted empirical cumulative process to Gaussian process, concentrated on continuous sample functions. Of course, both of these results are true under some regularity condition on the quantities involved.
Hira Lal Koul. "Some Convergence Theorems for Ranks and Weighted Empirical Cumulatives." Ann. Math. Statist. 41 (5) 1768 - 1773, October, 1970. https://doi.org/10.1214/aoms/1177696824