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 $E=[0,1]$.
Citation
James Kuelbs. Thomas Kurtz. Joel Zinn. "A CLT for empirical processes involving time-dependent data." Ann. Probab. 41 (2) 785 - 816, March 2013. https://doi.org/10.1214/11-AOP711
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