Discrepancy, chaining and subgaussian processes

Details Discrepancy, chaining and subgaussian processes Discrepancy, chaining and subgaussian processes

2011-04-08

arxiv.org/abs/1104.1508v1

Annals of Probability 2011, Vol. 39, No. 3, 985-1026

Mathematics

Probability

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

Scientific paper

10.1214/10-AOP575

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.

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