Explicit Universal Bounds for Cumulants via Moments
Jiechen Zhang
TL;DR
This work derives explicit, distribution-free bounds for the nth cumulant κ_n(X) in terms of the nth absolute moment M_n(X), with coefficients C_n that scale like (n−1)!/ρ^n and improve on classical n^n-growth bounds. It introduces a hierarchy of refinements—raw, centered, and symmetric—each yielding tighter constants through the moment–cumulant partition formula and Lyapunov-type inequalities, and it extends these bounds to joint cumulants. The coefficients are analyzed via generating functions and partition combinatorics, revealing precise asymptotics in terms of ordered Bell numbers and related structures; the bounds are strict for non-degenerate distributions but asymptotically efficient, yielding non-asymptotic constants for Edgeworth expansions and Bernstein-type concentration. The framework further connects to analytic properties of cumulant generating functions and demonstrates applications to CGF analyticity and tail bounds, with potential extensions to dependent data and high-dimensional statistics.
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 strict (not attainable by non-degenerate distributions), they are asymptotically efficient given the limited information of a single moment. The utility of the bounds is demonstrated through applications to analyticity of CGFs and concentration inequalities.