{"ID":2859245,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.05739","arxiv_id":"2510.05739","title":"Explicit Universal Bounds for Cumulants via Moments","abstract":"We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $κ_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\\lvertκ_n(X)\\rvert \\le C_n M_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n \\sim (n-1)!/ρ^{\\,n}$, which offers an exponential improvement over classical bounds where the coefficients grow superexponentially (on the order of $n^n$). We present a hierarchy of refinements where the rate parameter $ρ$ increases as the functional $M_n(X)$ incorporates more structural information. The most general bound uses the raw moment $M_n(X)=\\mathsf{E}[\\lvert X\\rvert^n]$ with rate $ρ=\\ln 2 \\approx 0.693$. Using the central moment $M_n(X)=\\mathsf{E}[\\lvert X-\\mathsf{E}[X]\\rvert^n]$ improves the rate to $ρ_{\\mathrm{cen}} \\approx 1.146$, while assuming symmetry yields even higher rates. The proof is elementary, combining the moment-cumulant partition formula with a uniform moment-product inequality. We further prove that while these bounds are not attainable whenever the relevant coefficient is positive, they are asymptotically efficient given the limited information of a single moment. The utility of the bounds is demonstrated through an application to standardized cumulants of independent sums.","short_abstract":"We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $κ_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\\lvertκ_n(X)\\rvert \\le C_n M_n(X)$. Our principal contribution is the derivation of coefficie...","url_abs":"https://arxiv.org/abs/2510.05739","url_pdf":"https://arxiv.org/pdf/2510.05739v3","authors":"[\"Jiechen Zhang\"]","published":"2025-10-07T10:00:02Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CO\",\"math.ST\"]","methods":"[]","has_code":false}