We consider empirical processes based on independent observations. We prove that the weak convergence of weighted empirical processes of i.i.d. random variables for the optimal class of weight function as proven in Csörgő, Csörgő, Horváth and Mason (Ann. Probabl. 14, 1986, 31-85) continueds to hold under contiguous measures for the same class of weight functions.

Brownian bridge, contiguity, Empirical process, weak convergence, weight functions
Statistics and Probability Letters
School of Mathematics and Statistics

Szyszkowicz, B. (1994). The Chibisov-O'Reilly theorem for empirical processes under contiguous measures. Statistics and Probability Letters, 19(2), 153–159. doi:10.1016/0167-7152(94)90149-X