A CLT for empirical processes involving time-dependent data

Abstract: For stochastic processes ${X_t:t\in E}$, we establish sufficient conditions for the empirical process based on ${I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. 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 $E=[0,1]$.