{"ID":3004633,"CreatedAt":"2026-06-03T03:09:48.883664427Z","UpdatedAt":"2026-06-05T11:43:53.432517148Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.03973","arxiv_id":"2606.03973","title":"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 $X \\sim μ$ 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} \u003c \\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","short_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} \u003c \\infty, \\]...","url_abs":"https://arxiv.org/abs/2606.03973","url_pdf":"https://arxiv.org/pdf/2606.03973v1","authors":"[\"Reese Pathak\",\"Nikita Zhivotovskiy\"]","published":"2026-06-02T17:55:38Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"cs.IT\",\"math.ST\"]","methods":"[]","has_code":false}