A remark on the majorizing measures theorem for general processes
2026-06-02 • Information Theory
Information Theory
AI summaryⓘ
The authors show that a certain inequality known as the lower bound in the majorizing measures theorem applies not just to Gaussian random vectors but to a wide class of centered random vectors with finite KL divergence scaling. They prove that the expected maximum projection of such a vector onto a set T is at least proportional to a chaining functional called γ₂(T). This generalizes a key result for Gaussian processes and uses a tool called the rate-distortion integral. Their work connects information theory concepts to probability bounds on random vectors.
majorizing measures theoremcentered random vectorKullback-Leibler divergenceTalagrand's generic chainingrate-distortion integralGaussian processesrandom vectorsexpected supremummeasure concentrationinformation theory
Authors
Reese Pathak, Nikita Zhivotovskiy
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