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January, 1987 Measurability Problems for Empirical Processes
Michel Talagrand
Ann. Probab. 15(1): 204-212 (January, 1987). DOI: 10.1214/aop/1176992264


To a class $\mathscr{F}$ of bounded functions on a probability space we associate two classes $\mathscr{F}_r$ and $\mathscr{F}_s$. The class $\mathscr{F}$ is a Donsker class if and only if $\mathscr{F}_r$ and $\mathscr{F}_s$ are Donsker classes. The class $\mathscr{F}_r$ corresponds to a separable version of the empirical process. It is obtained by applying a special type of lifting to $\mathscr{F}$. The class $\mathscr{F}_s$ consists of positive functions that are zero almost surely. It concentrates the pathology of $\mathscr{F}$ with respect to measurability. We use this method to prove without any measurability assumption a general contraction principle for processes that satisfy the central limit theorem.


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Michel Talagrand. "Measurability Problems for Empirical Processes." Ann. Probab. 15 (1) 204 - 212, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0622.60040
MathSciNet: MR877598
Digital Object Identifier: 10.1214/aop/1176992264

Primary: 60G05
Secondary: 28A51 , 60F05

Keywords: contraction principle , Donsker class , lifting

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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