The Annals of Probability

A CLT for empirical processes involving time-dependent data

James Kuelbs, Thomas Kurtz, and Joel Zinn

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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 $E=[0,1]$.

Article information

Ann. Probab., Volume 41, Number 2 (2013), 785-816.

First available in Project Euclid: 8 March 2013

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Central limit theorems empirical processes


Kuelbs, James; Kurtz, Thomas; Zinn, Joel. A CLT for empirical processes involving time-dependent data. Ann. Probab. 41 (2013), no. 2, 785--816. doi:10.1214/11-AOP711.

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