Explicit Universal Bounds for Cumulants via Moments

Abstract

We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $\kappa_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\lvert\kappa_n(X)\rvert \le C_n M_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n \sim (n-1)!/\rho^{\,n}$, 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 $\rho$ 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 $\rho=\ln 2 \approx 0.693$. Using the central moment $M_n(X)=\mathsf{E}[\lvert X-\mathsf{E}[X]\rvert^n]$ improves the rate to $\rho_{\mathrm{cen}} \approx 1.146$, 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.