The Annals of Probability

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

Kenneth S. Alexander

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Abstract

Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class $\mathscr{F}$ of functions are proved under several kinds of conditions on $\mathscr{F}$. 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.

Article information

Source
Ann. Probab., Volume 12, Number 4 (1984), 1041-1067.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993141

Digital Object Identifier
doi:10.1214/aop/1176993141

Mathematical Reviews number (MathSciNet)
MR757769

Zentralblatt MATH identifier
0549.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60F15: Strong theorems 60G57: Random measures

Keywords
Empirical process exponential bound law of the iterated logarithm Vapnik-Cervonenkis class metric entropy

Citation

Alexander, Kenneth S. Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm. Ann. Probab. 12 (1984), no. 4, 1041--1067. doi:10.1214/aop/1176993141. https://projecteuclid.org/euclid.aop/1176993141


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Corrections

  • See Correction: Kenneth S. Alexander. Correction: Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm. Ann. Probab., Volume 15, Number 1 (1987), 428--430.