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Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics
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  • Published: September 1988

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

  • R. Helmers1,
  • P. Janssen2 &
  • R. Serfling3ย 

Probability Theory and Related Fields volumeย 79,ย pages 75โ€“93 (1988)Cite this article

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  • 38 Citations

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Summary

We study a nonclassical form of empirical df H nwhich is of U-statistic structure and extend to H nthe classical exponential probability inequalities and Glivenko-Cantelli convergence properties known for the usual empirical df. An important class of statistics is given byT(H n), where T(ยท) 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.

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Author information

Authors and Affiliations

  1. Centre for Mathematics and Computer Science, Kruislaan 413, SJ-1098, Amsterdam, The Netherlands

    R. Helmers

  2. Limburgs Universitair Centrum, B-3610, Diepenbeek, Belgium

    P. Janssen

  3. Department of Mathematical Sciences, Johns Hopkins University, 21218, Baltimore, MD, USA

    R. Serfling

Authors
  1. R. Helmers
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  2. P. Janssen
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  3. R. Serfling
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Additional information

Research supported by the U.S. Department of Navy under Office of Naval Research Contract No. N00014-79-C-0801 and by NATO under Research Grant No. 0034/87

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Helmers, R., Janssen, P. & Serfling, R. Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics. Probab. Th. Rel. Fields 79, 75โ€“93 (1988). https://doi.org/10.1007/BF00319105

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  • Received: 24 February 1986

  • Revised: 05 March 1988

  • Issue Date: September 1988

  • DOI: https://doi.org/10.1007/BF00319105

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Important Class
  • Strong Convergence
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