Journal of Theoretical Probability

, Volume 4, Issue 3, pp 485–510 | Cite as

Uniform and universal Glivenko-Cantelli classes

  • R. M. Dudley
  • E. Giné
  • J. Zinn
Article

Abstract

A class\(\widetilde{F}\) of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measuresP n converge to the trueP uniformly over\(\widetilde{F}\) almost surely.\(\widetilde{F}\) is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli Cantelli class for all lawsP on a measurable space, and a uniform Glivenko-Cantelli class if the convergence is also uniform inP. We give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.

Key Words

Laws of large numbers Vapnik-Červonenkis classes 

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Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • R. M. Dudley
    • 1
  • E. Giné
    • 2
  • J. Zinn
    • 3
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridge
  2. 2.Department of MathematicsUniversity of ConnecticutStorrs
  3. 3.Department of MathematicsTexas A & M UniversityCollege Station

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