Symmetrization approach to concentration inequalities for empirical processes

Details Symmetrization approach to concentration inequalities for empirical processes Symmetrization approach to concentration inequalities for empirical processes

2004-05-18

arxiv.org/abs/math/0405354v1

2003 Ann. Probab. 31 No.4

Mathematics

Probability

15 pages

Scientific paper

We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube {-1,+1}^n. As another application of this technique we prove new Vapnik-Chervonenkis type inequalities. For example, for VC-classes of functions we prove a classical inequality of Vapnik and Chervonenkis only with normalization by the sum of variance and sample variance.

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