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 $X\sim \hat{I}¼$ 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 $\hat{I}{¼}_{\hat{I}}¸$ denotes the law of the translate $\hat{I}¸+X$. 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.