Journal of Theoretical Probability

, Volume 8, Issue 4, pp 825–831 | Cite as

On the reversed sub-martingale property of empirical discrepancies in arbitrary sample spaces

  • Franz Strobl
Article
  • 33 Downloads

Abstract

The empirical discrepancy is defined as a supremum over a class of functions of a collection of centered sample averages. For uncountable classes the discrepancy need not be measurable, and distributional assertions can become dependent on the structure of the underlying probability space. This paper shows that one such assertion—the reversed sub-martingale property—is valid when interpreted in terms of measurable cover functions for the canonical model, but that it can fail in other constructions of the underlying model.

Key Words

Empirical discrepancies reversed sub-martingale property canonical model measurable cover function 

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References

  1. 1.
    Dudley, R. M. (1984). A course on empirical processes. École d'été de probabilités de Saint-Flour XII-1982.Lecture Notes in Math. 1097, 1–142.MATHMathSciNetGoogle Scholar
  2. 2.
    Dudley, R. M., and Philipp, W. (1983). Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes.Z. Wahrsch. verw. Gebiete 62, 509–552.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Gaenssler, P. (1983).Empirical Processes. IMS Lecture Notes—Monograph Series, Vol. 3, Institute of Mathematical Statistics, Hayward, California.Google Scholar
  4. 4.
    Halmos, P. R. (1950).Measure Theory, Van Nostrand, Princeton.Google Scholar
  5. 5.
    Pollard, D. (1984).Convergence of Stochastic Processes, Springer, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Franz Strobl
    • 1
  1. 1.Mathematical InstituteUniversity of MunichMunichGermany

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