Upper bounds for the L^q empirical process via generic chaining

TL;DR

This paper uses generic chaining to establish upper bounds for the L^q empirical process of sub-Gaussian classes for 1 ≤ q < ∞, resolving an open problem and impacting Banach space geometry.

Contribution

It extends upper bounds for the L^q process to all q ≥ 1 using generic chaining, solving a previously open problem.

Findings

Derived upper bounds for L^q processes for 1 ≤ q ≤ 2.

Extended bounds to all q ≥ 1 combining previous results.

Applied bounds to Banach space geometry and convex body sections.

Abstract

Using the generic chaining method, we derive upper bounds for the \(L^q\) process of sub-Gaussian classes when \(1 \le q \le 2\), thereby resolving an open problem posed by Al-Ghattas, Chen, and Sanz-Alonso in arXiv:2502.16916. Combined with the results of arXiv:2502.16916, this yields upper bounds for the \(L^q\) process for all \(1 \le q < \infty\). We also present corollaries of this result in the geometry of Banach spaces, including high-probability bounds on the \(\ell_q\) norm diameter of random hyperplane sections of convex bodies where the subspaces are not necessarily uniformly distributed on the Grassmannian manifold and the restricted isomorphic property for \(\ell_q\) norm.