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.