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 $E=[0,1]$.