Probability Theory and Related Fields

, Volume 79, Issue 1, pp 75–93

Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics

  • R. Helmers
  • P. Janssen
  • R. Serfling
Article

DOI: 10.1007/BF00319105

Cite this article as:
Helmers, R., Janssen, P. & Serfling, R. Probab. Th. Rel. Fields (1988) 79: 75. doi:10.1007/BF00319105

Summary

We study a nonclassical form of empirical df Hnwhich is of U-statistic structure and extend to Hnthe classical exponential probability inequalities and Glivenko-Cantelli convergence properties known for the usual empirical df. An important class of statistics is given byT(Hn), where T(·) is a generalized form of L-functional. For such statisticswe prove almost sure convergence using an approach which separates the functional-analytic and stochastic components of the problem and handles the latter component by application of Glivenko-Cantelli type properties.Classical results for U-statistics and L-statistics are obtained as special cases without addition of unnecessary restrictions.Many important new types of statistics of current interest are covered as well by our result.

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • R. Helmers
    • 1
  • P. Janssen
    • 2
  • R. Serfling
    • 3
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands
  2. 2.Limburgs Universitair CentrumDiepenbeekBelgium
  3. 3.Department of Mathematical SciencesJohns Hopkins UniversityBaltimoreUSA