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Probability inequalities for empirical processes and a law of the iterated logarithm. (English) Zbl 0549.60024
Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables taking values in a space (X,\({\mathcal A})\). Denote \(\nu_ n(f)=\sum^{n}_{j=1}(f(X_ j)-Ef(X_ j))/\sqrt{n}\), \(f\in {\mathcal F}\), the stochastic process indexed by a class \({\mathcal F}\) of functions on X. Sharp exponential bounds for the probabilities of deviations and laws of the iterated logarithm for \(\sup_{{\mathcal F}}|\nu_ n(f)|\) are proved for some uniformly bounded \({\mathcal F}\). The bounds for these probabilities are used to obtain rates of convergence in total variation for empirical processes on the integers.
Reviewer: V.M.Kruglov

60F10 Large deviations
60F15 Strong limit theorems
60G57 Random measures
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