A CLT for empirical processes involving time-dependent data

For stochastic processes {Xt : t∈E}, we establish sufficient conditions for the empirical process based on {IXt≤y−Pr(Xt≤y) : t∈E,y∈R} to satisfy the CLT uniformly in t∈E, y∈R. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and E=[0,1].