Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm

Given a class -F of functions on X, we can view Vn as a stochastic process indexed by -F and consider limit theorems for this process. To prove such theorems it is often helpful to have bounds on the tail of the r.v. sup _W I n(f ) I (see for example Dudley, 1978, Kuelbs and Dudley, 1980, Dudley and Phillip, 1983, or Gine and Zinn, 1983.) Our main question of interest is, for what classes -F can "best possible" bounds be obtained? So first we must ask, what are these best possible bounds? Of course

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