A remark on the majorizing measures theorem for general processes

Abstract

We show that the lower bound in the majorizing measures theorem holds for a large class of random vectors. Specifically, suppose $X \sim \mu$ is a centered random vector in $\mathbf{R}^n$ with \[ C_{\mathrm{KL}}(\mu) = \sup_{\substack{\theta \neq \eta \\ \theta, \eta \in \mathbf{R}^n}} \frac{\mathrm{KL}(\mu_\theta \| \mu_\eta)}{\|\theta - \eta\|_2^2} < \infty, \] where $\mu_\theta$ denotes the law of the translate $\theta + X$. Then, for every nonempty, bounded $T \subset \mathbf{R}^n$, \[ \sqrt{C_{\mathrm{KL}}(\mu)}\, \mathbf{E}_\mu \Big[\sup_{t \in T} \, \langle X, t \rangle \Big] \gtrsim \gamma_2(T), \] 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.