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

Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class F of functions are proved under several kinds of conditions on F. These bounds are used to establish laws of the iterated logarithm for this supremum and to obtain rates of convergence in total variation for empirical processes on the integers.