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
Recommended Citation
Rakhlin, A., Sridharan, K., & Tewari, A. (2015). Sequential Complexities and Uniform Martingale Laws of Large Numbers. Probability Theory and Related Fields, 161 (1), 111-153. http://dx.doi.org/10.1007/s00440-013-0545-5
Date Posted: 27 November 2017
This document has been peer reviewed.