We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs inf(εi) supf∈F|∑i=1k εif(Xi)| 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 ℝk 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.
"Discrepancy, chaining and subgaussian processes." Ann. Probab. 39 (3) 985 - 1026, May 2011. https://doi.org/10.1214/10-AOP575