Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics
- Cite this article as:
- Helmers, R., Janssen, P. & Serfling, R. Probab. Th. Rel. Fields (1988) 79: 75. doi:10.1007/BF00319105
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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.