Explicit Universal Bounds for Cumulants via Moments

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 $\mid {\kappa }_n(X)\mid \le C_n{M}_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n\sim (n-1)!\mathrm{/}{\rho }^n$, 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 $M_n(X)=\mathsf{E}[\mid X{\mid }^n]$ with rate $\rho =\ln 2\approx 0.693$. Using the central moment $M_n(X)=\mathsf{E}[\mid X-\mathsf{E}[X]{\mid }^n]$ improves the rate to ${\rho }_{\mathrm{c}\mathrm{e}\mathrm{n}}\approx 1.146$, 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.