Discrepancy, chaining and subgaussian processes

Abstract

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs ${\inf }_{({\epsilon }_i)}{\sup }_{f\in F}|{\sum }_{i=1}^k{\epsilon }_{i}f(X_i)|$ is asymptotically smaller than the expectation over signs as a function of the dimension $k$, if the canonical Gaussian process indexed by $F$ is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of ${\mathbb{R}}^k$ using properties of the canonical Gaussian process the set indexes, and then obtain quantitative structural information on a typical coordinate projection of a subgaussian class.

Published in at http://dx.doi.org/10.1214/10-AOP575 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• A probabilistic approach to the geometry of the \ell_p^n-ball
• Reconstruction and subgaussian operators