Squared-norm empirical processes

Abstract

This note extends a result of Mendelson on the supremum of a quadratic process to squared norms of functions taking values in a Banach space. Our method of proof is a reduction by a symmetrization argument and simple observation about the additivity of the generic chaining functional. We demonstrate an application to positive linear functionals of the sample covariance matrix and the apparent variance explained by principal components analysis (PCA).