Necessary and sufficient conditions for the weak convergence in stronger than the usual metrics of empirical, inverse empirical and partial sum processes are obtained. The results are related to work of Chibisov and of Pyke and Shorack.
"On the Weak Convergence of Empirical Processes in Sup-norm Metrics." Ann. Probab. 2 (4) 642 - 651, August, 1974. https://doi.org/10.1214/aop/1176996610