A Universal Moments-Only Bound for Cumulants

Abstract: We establish a simple, universal inequality that bounds the $n$-th cumulant of a real-valued random variable using only its $n$-th (absolute or central) moment. Specifically, for any integer $n \ge 1$, the $n$-th cumulant $\kappa_n(X)$ satisfies [ \lvert \kappa_n(X) \rvert \;\le\; C_n\, \mathbb{E}\lvert X-\mathbb{E}X\rvert^{\,n}, ] with an alternative bound in terms of $\mathbb{E}\lvert X\rvert^{\,n}$ in the uncentered form. The coefficient $C_n$ is derived from the combinatorial structure of the moment--cumulant formula and exhibits the asymptotic behavior $C_n \sim (n-1)!/\rho^{\,n}$, giving an exponential improvement over classical bounds that grow on the order of $n^n$. In full generality, the bound involves the ordered Bell numbers, corresponding to a rate parameter $\rho=\ln 2\approx 0.693$. For $n\ge 2$, shift-invariance of cumulants yields a universal centered refinement with parameter $\rho_0\approx 1.146$, determined by $e^{\rho_0}=2+\rho_0$. For symmetric random variables, the bound sharpens further to $\rho_{\mathrm{sym}}=\operatorname{arcosh}2\approx 1.317$. These results extend naturally to the multivariate setting, providing uniform control of joint cumulants under the same minimal moment assumptions.