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.

翻译：我们证明，对于一大类随机向量，主测度定理中的下界成立。具体地，设 $X \sim μ$ 是 $\mathbf{R}^n$ 中的中心化随机向量，满足 \[ C_{\mathrm{KL}}(μ) = \sup_{\substack{θ\neq η\\ θ, η\in \mathbf{R}^n}} \frac{\mathrm{KL}(μ_θ\| μ_η)}{\|θ- η\|_2^2} < \infty, \] 其中 $μ_θ$ 表示平移 $θ+ X$ 的分布。那么，对于任意非空有界集合 $T \subset \mathbf{R}^n$，有 \[ \sqrt{C_{\mathrm{KL}}(μ)}\, \mathbf{E}_μ\Big[\sup_{t \in T} \, \langle X, t \rangle \Big] \gtrsim γ_2(T), \] 这里右侧表示Talagrand的通用链条泛函。作为特例，该结果恢复了中心化高斯过程主测度定理中的下界。我们的论证关键依赖于J. Liu近期引入的率失真积分。