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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

##### MSC:
 60F10 Large deviations 60F15 Strong limit theorems 60G57 Random measures
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