A Characterization of the Cumulants as Continuous Moment-Based Statistics

Abstract: 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.

• The paper proves that any continuous, additive statistic based on finitely many moments is necessarily a linear combination of cumulants.
• It employs an algebraic approach using the Hurwitz product and a polynomial change of coordinates via Bell polynomials.
• The result reinforces cumulants' canonical role in distribution theory, impacting statistical inference and asymptotic analysis.

Characterization of Cumulants as Continuous Moment-Based Statistics

Overview

The paper "A Characterization of the Cumulants as Continuous Moment-Based Statistics" (2606.29615) rigorously addresses the inverse problem of cumulant additivity: Given a statistic that is both a continuous function of finitely many moments and additive under the sum of independent random variables, is it necessarily a linear combination of cumulants? The author affirms this, providing an elementary proof based on an algebraic interpretation of convolution (the Hurwitz product), and a linearizing change of coordinates via polynomial transformations related to the Bell polynomials.

This theorem has broad implications for the theory of statistical functionals, situating the classical cumulants as the unique family of continuous, additive, moment-based statistics.

Cumulants and Their Additivity Property

Cumulants $\kappa_n$ are defined, for random variables with finite $n$-th moments, through the expansion of the cumulant generating function:

$$\log\left(E[e^{zX}]\right) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!} \kappa_\ell(X) + o(z^n),\quad z\to 0,\,z\in i\mathbb{R}$$

Their primary property, distinguishing them among moment-based functionals, is their strict additivity under convolution: For $X,Y$ independent, $\kappa_n(X+Y) = \kappa_n(X) + \kappa_n(Y)$ for all $n$. This property, which is fundamental in limit theory and related applications, underpins the identification result developed in the paper.

Lower order cumulants coincide with familiar statistics—for instance, $\kappa_1$ (mean), $\kappa_2$ (variance)—while higher order ones measure skewness, kurtosis, and further deviations from normality. The vanishing of all cumulants of order $\geq 3$ characterizes the normal distribution.

Main Theorem: Moment-Based Additive Statistics

The central result is:

If $F:L^n\to\mathbb{R}$ is a statistic depending continuously on finitely many moments (i.e., $n$0), and $n$1 is additive under convolution, then $n$2 must be a linear combination of cumulants up to degree $n$3.

The hypothesis requires only continuity in finitely many moments and additivity for independent sums. The conclusion is sharply rigid: Any such functional is explicitly given by linear combinations of cumulants. The polynomial representations arise via logarithmic Bell polynomials, leveraging the combinatorics of Faà di Bruno's formula and exponential generating functions.

Of note, the result generalizes prior characterizations—those requiring the functional to be a polynomial in the moments (Thiele) and those assuming additional topological structure (Mattner). The author’s approach is distinctively elementary, relying on a direct algebraic translation of the convolution operation to a Hurwitz product on finite-dimensional moment space.

Algebraic Reformulation and Proof Sketch

The proof reduces the functional equation

$n$4

to an algebraic system on the space of moment sequences $n$5, with convolution corresponding to the Hurwitz product $n$6:

$n$7

A change of basis $n$8 is constructed, defined polynomially in the moments, such that $n$9 transforms $$\log\left(E[e^{zX}]\right) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!} \kappa_\ell(X) + o(z^n),\quad z\to 0,\,z\in i\mathbb{R}$$0 to ordinary addition, allowing the Cauchy functional equation to be invoked for the continuous, additive function $$\log\left(E[e^{zX}]\right) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!} \kappa_\ell(X) + o(z^n),\quad z\to 0,\,z\in i\mathbb{R}$$1. The solution space is shown to be finite-dimensional and precisely spanned by cumulants.

A notable computational detail is the explicit representation of cumulants as logarithmic Bell polynomials in the moments, with the triangular form of the change-of-basis matrix ensuring linear independence.

Context and Related Work

Historically, the quest to characterize cumulants uniquely among moment-based statistics derives from Thiele's investigations in the 19th century. Thiele and subsequent commentators (Rota) highlighted properties such as translation invariance, additivity for independent sums, and polynomial dependence on moments. Mattner later established a functional-analytic generalization: Under a suitable topology, any continuous, additive statistic on the space of probability measures with all moments finite is a linear combination of cumulants. The result here, achieved via elementary algebra and combinatorics, interpolates between Thiele’s classical approach and Mattner’s topological generality.

Implications for Probability, Statistics, and Theoretical Developments

This theorem clarifies the extremal status of cumulants among all continuous, additive, moment-based statistics, directly informing the canonical choice of these statistics in distribution theory, asymptotic analysis, and the algebraic study of convolution semigroups. The result restricts the set of potential statistics with desirable properties, shaping how generalizations (e.g., in non-commutative probability, free probability) might proceed.

The formulation via the Hurwitz product and moment space topology suggests further generalizations, such as to infinite-dimensional moment sequences, non-commutative algebras, or generalized cumulant concepts in quantum probability. Additionally, it supports the methodological use of cumulants in limit theorems, statistical inference, and the study of deviation from normality, by ensuring maximality of cumulant statistics under weak structural requirements.

Numerical and Theoretical Strength

The main claim is structurally strong: Every continuous statistic additive for the sum of independent variables, and depending only on finitely many moments, must be a linear combination of cumulants. The argument is constructive, providing explicit bases and transformations, and applies under minimal regularity assumptions on the functionals. There are no contradictory claims; all conclusions are rigorously derived from established functional and algebraic theory.

Conclusion

By connecting moment-based additivity, continuity, and the algebraic structure of convolution, this paper rigorously demonstrates the rigidity of cumulant statistics in the finite moment space setting. The technical innovation, invoking the Hurwitz product and an explicit linearization, yields a transparent and constructive proof of an important structural result about statistics on random variables. This work clarifies the theoretical foundation of cumulants and points towards further studies in algebraic and functional characterizations of distributional invariants.