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Entropy conditions for L-r-convergence of empirical processes

Caponnetto, A; De Vito, E; Pontil, M; (2009) Entropy conditions for L-r-convergence of empirical processes. ADV COMPUT MATH , 30 (4) 355 - 373. 10.1007/s10444-008-9072-9.

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Abstract

The law of large numbers (LLN) over classes of functions is a classical topic of empirical processes theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko-Cantelli classes) have been widely studied in the literature. An elegant sufficient condition for such a property is finiteness of the Koltchinskii-Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities ( e. g. the Vapnik-Chervonenkis dimension). In this paper, we endow the class of functions F with a probability measure and consider the LLN relative to the associated L-r metric. This framework extends the case of uniform convergence over F, which is recovered when r goes to infinity. The main result is a L-r-LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii-Pollard entropy integral.

Type:Article
Title:Entropy conditions for L-r-convergence of empirical processes
DOI:10.1007/s10444-008-9072-9
Keywords:Empirical processes, Uniform entropy, Rademacher averages, Glivenko-Cantelli classes
UCL classification:UCL > School of BEAMS > Faculty of Engineering Science > Computer Science

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