Date of this Version
Probability Theory and Related Fields
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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-013-0545-5.
empirical processes, dependent data, uniform Glivenko-Cantelli classes, rademacher averages, sequential prediction
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.