Statistics Papers

Document Type

Journal Article

Date of this Version

2-2015

Publication Source

Probability Theory and Related Fields

Volume

161

Issue

1

Start Page

111

Last Page

153

DOI

10.1007/s00440-013-0545-5

Abstract

We establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide “sequential” (non-i.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data. As a direct application of our results, we provide exponential inequalities for sums of martingale differences in Banach spaces.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-013-0545-5.

Keywords

empirical processes, dependent data, uniform Glivenko-Cantelli classes, rademacher averages, sequential prediction

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Date Posted: 27 November 2017

This document has been peer reviewed.