A remark on the majorizing measures theorem for general processes

2026-06-02

AI Summary

This work extends the lower bound results of the majorizing measure theorem from Gaussian processes to classes of centered random vectors satisfying a finite Kullback–Leibler (KL) divergence condition. By integrating Talagrand’s γ₂ functional, KL-divergence-based analysis of random vectors, and Liu’s rate–distortion integral approach, the authors establish a universal lower bound relating the expected supremum of linear functionals over such random vectors to the γ₂ functional. This result not only recovers the Gaussian case as a special instance but also substantially broadens the applicability of majorizing measure lower bounds, highlighting the central role of the γ₂ functional in far more general non-Gaussian settings.

Abstract

We show that the lower bound in the majorizing measures theorem holds for a large class of random vectors. Specifically, suppose $X \sim μ$ is a centered random vector in $\mathbf{R}^n$ with \[ C_{\mathrm{KL}}(μ) = \sup_{\substack{θ\neq η\\ θ, η\in \mathbf{R}^n}} \frac{\mathrm{KL}(μ_θ\| μ_η)}{\|θ- η\|_2^2} < \infty, \] where $μ_θ$ denotes the law of the translate $θ+ X$. Then, for every nonempty, bounded $T \subset \mathbf{R}^n$, \[ \sqrt{C_{\mathrm{KL}}(μ)}\, \mathbf{E}_μ\Big[\sup_{t \in T} \, \langle X, t \rangle \Big] \gtrsim γ_2(T), \] where the righthand side denotes Talagrand's generic chaining functional. This result recovers, as a special case, the lower bound in the majorizing measures theorem for centered Gaussian processes. Our argument critically relies on the rate-distortion integral, recently introduced by J. Liu

Problem

• majorizing measures theorem
• random vectors
• Kullback-Leibler divergence
• generic chaining
• lower bound

Innovation

• majorizing measures theorem
• rate-distortion integral
• Kullback-Leibler divergence
• generic chaining
• random processes