A Characterization of the Cumulants as Continuous Moment-Based Statistics

Cumulants are classical statistics associated with a random variable, defined as polynomial functions of its moments and distinguished by their additivity under convolution of distributions. A statistic is the name given to a function of a random variable, and a moment-based statistic is one that depends only on the moments $(\mathbb{E}[X^n]{)}_{n\in \mathbb{N}}$. We prove a converse: any statistic depending continuously on finitely many moments and additive for independent sums must be a linear combination of cumulants. The proof uses an algebraic reformulation of the problem via the Hurwitz product and a linearizing change of coordinates. This result also follows from the more general theorem of Mattner \cite{mattner}, but our approach is elementary and self-contained.