Probability Theory and Related Fields

, Volume 79, Issue 1, pp 75-93

First online:

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

  • R. HelmersAffiliated withCentre for Mathematics and Computer Science
  • , P. JanssenAffiliated withLimburgs Universitair Centrum
  • , R. SerflingAffiliated withDepartment of Mathematical Sciences, Johns Hopkins University

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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(·) 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.