Extensions of Billingsley's theorems on weak convergence of empirical processes

Abstract

In [4], Sen proved Theorems 22.1 and 22.2 of Billingsley [1] on the weak convergence of empirical distribution functions for sequences of Φ-mixing random variables to appropriate Gaussian random functions under less stringent regularity conditions. In this note, we shall prove the same theorems under weaker conditions than Sen's ones.