A class of empirical processes having the structure of $U$-statistics is considered. The weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology. As an application, a central limit theorem is derived for a very general class of non-parametric statistics.
"Convergence of a Class of Empirical Distribution Functions of Dependent Random Variables." Ann. Probab. 11 (3) 745 - 751, August, 1983. https://doi.org/10.1214/aop/1176993518