, Volume 79, Issue 1, pp 75-93

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

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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.

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