Upper bounds for the L^q empirical process via generic chaining
Zong Shang
TL;DR
This work derives sharp, high-probability upper bounds for the L^q empirical process of sub-Gaussian classes with 1 ≤ q ≤ 2 by employing a refined generic chaining analysis. The bound depends on the γ_2(F, d_{ψ_2}) complexity and the diameter diam(F, d_{ψ_2}), with an explicit phase transition at q = 2 reflecting the mixed Gaussian/Weibull tail behavior of |f|^q. The results unify the analysis for 1 ≤ q < ∞ with prior q ≥ 2 results, and yield important corollaries in Banach space geometry, including RIP-type control and ell_p-diameter bounds for random sections of convex bodies, as well as a Dvoretzky–Milman-type phenomenon. The methods leverage a two-segment chaining (Gaussian-initial and Weibull-terminal) and a combination of stochastic and deterministic arguments to achieve tight, interpretable dependence on γ_2 and diam, with clear implications for high-dimensional probability and geometric functional analysis.
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.