Cumulants: Definitions, Properties, Applications

• Cumulants are polynomial functions of moments that linearize the additive effect of independent random variables.
• They provide a natural coordinate system for convolution, enabling efficient computation and analysis in fields like combinatorics, algebraic statistics, and quantum theory.
• Applications range from characterizing statistical distributions to modeling fluctuations in heavy-ion collisions, random matrices, and related physical phenomena.

Cumulants are classical statistics associated with a random variable, defined as polynomial functions of its moments and distinguished by additivity under convolution of distributions. For a random variable $X$ with finite moments, the $n$-th cumulant $\kappa_n(X)$ is obtained from the logarithm of the moment generating function near $0$, so cumulants are coefficients in the Taylor expansion of $\log(E[e^{zX}])$. This construction makes cumulants polynomial functions of moments, with the inverse relation expressing moments as polynomials in cumulants; it also makes cumulants the natural “linear coordinates” for convolution, because they add for independent sums (Cerda, 28 Jun 2026). In modern probability and adjacent fields, the notion has been extended to free, Boolean, monotone, finite free, rectangular, and second-order settings, and has become a common organizing principle in combinatorics, algebraic statistics, random matrices, and quantum field theory (Ebrahimi-Fard et al., 2014).

1. Classical definition and elementary structure

For a random variable $X$ with finite moments, the paper "A Characterization of the Cumulants as Continuous Moment-Based Statistics" defines cumulants by

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

In this sense, cumulants are the coefficients of the logarithm of the moment generating function near the origin (Cerda, 28 Jun 2026).

The first cumulants are explicitly

$\kappa_1(X)=E[X],$

$\kappa_2(X)=E[X^2]-E[X]^2,$

$\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$

$n$0

Accordingly, $n$1 is the mean and $n$2 is the variance. Higher cumulants measure increasingly subtle departures from Gaussian behavior; for a normal distribution, all cumulants of order $n$3 and higher vanish (Cerda, 28 Jun 2026).

A complementary formulation, used in heavy-ion phenomenology, identifies the lowest cumulants with familiar centered fluctuation measures. For baryon number $n$4, the first three satisfy $n$5 = mean, $n$6 = variance, and $n$7 is the skewness-related third cumulant; thermodynamically, the same quantities appear as derivatives of the pressure with respect to baryon chemical potential (Sorensen et al., 2021). This dual statistical and thermodynamic role is one reason cumulants recur across disparate areas.

2. Moment–cumulant relations and combinatorial formulas

The classical moment–cumulant correspondence is polynomial in both directions. Using incomplete Bell polynomials $n$8, moments can be written as

$n$9

while cumulants can be recovered from moments through

$\kappa_n(X)$0

Equivalently, the logarithmic Bell polynomials $\kappa_n(X)$1 appear in the formal identity

$\kappa_n(X)$2

which is precisely the polynomial formula expressing cumulants in terms of moments (Cerda, 28 Jun 2026).

A partition-theoretic version of the same relation writes the classical cumulant as

$\kappa_n(X)$3

with $\kappa_n(X)$4 and $\kappa_n(X)$5 the lattice of set partitions. This partition formula is the basis for exact coefficient bounds and for many generalizations (Zhang, 7 Oct 2025).

The combinatorics becomes especially vivid for the $\kappa_n(X)$6-semicircular law. Its even moments satisfy

$\kappa_n(X)$7

where $\kappa_n(X)$8 is the set of matchings and $\kappa_n(X)$9 counts crossings. Free cumulants restrict this enumeration to connected matchings, and classical cumulants obey the sharper formula

$0$0

where $0$1 is the crossing graph of the matching and $0$2 is the Tutte polynomial. Thus moments, free cumulants, and classical cumulants correspond to progressively more structured connected objects (Josuat-Vergès, 2012).

These formulas show that cumulants are not merely alternative coordinates. They encode the same moment data through Möbius inversion, Bell polynomials, or graph-weighted connected structures, depending on the ambient combinatorics.

3. Additivity, convolution, and characterization

The structural property that distinguishes classical cumulants is additivity under independent sums: $0$3 This follows from

$0$4

so the logarithm converts multiplication into addition (Cerda, 28 Jun 2026).

The 2026 characterization theorem makes this property essentially definitive. If

$0$5

for some continuous $0$6, and if $0$7 is additive for independent sums, then

$0$8

Equivalently, $0$9 is a linear combination of logarithmic Bell polynomials. In other words, among continuous statistics depending only on finitely many moments and additive under independent summation, cumulants form the complete basis (Cerda, 28 Jun 2026).

The proof reformulates moment addition via the Hurwitz product. If

$\log(E[e^{zX}])$0

then the moment sequence of $\log(E[e^{zX}])$1 is

$\log(E[e^{zX}])$2

with $\log(E[e^{zX}])$3. A change of coordinates $\log(E[e^{zX}])$4 then converts the Hurwitz product into ordinary addition: $\log(E[e^{zX}])$5 After composition with $\log(E[e^{zX}])$6, the functional equation becomes a continuous Cauchy equation, so linearity follows, and the cumulant polynomials emerge as the corresponding linear basis (Cerda, 28 Jun 2026).

This result also follows from a more general theorem of Mattner, but the cited proof is elementary and self-contained. Conceptually, it establishes that the canonical status of cumulants is not incidental: additivity plus finite moment dependence already forces them.

4. Free, Boolean, monotone, and second-order cumulants

In noncommutative probability, cumulants depend on the notion of independence. Free cumulants are indexed by noncrossing partitions, Boolean cumulants by interval partitions, and monotone cumulants by monotone or tree-weighted noncrossing structures. In the Hopf-algebraic shuffle framework, moments are encoded by characters and cumulants by infinitesimal characters; free cumulants are obtained from the left half-shuffle fixed-point equation

$\log(E[e^{zX}])$7

Boolean cumulants from

$\log(E[e^{zX}])$8

and monotone cumulants from the shuffle logarithm

$\log(E[e^{zX}])$9

The associated moment formulas are

$X$0

with monotone cumulants carrying the additional tree-factorial weights (Ebrahimi-Fard et al., 2017).

A related commutative/noncommutative unification interprets classical cumulants as the commutative specialization of the same half-shuffle mechanism. In that setting, classical cumulants satisfy $X$1, while free cumulants arise from the noncommutative fixed-point equation $X$2. The distinction between all partitions and noncrossing partitions is thereby traced to the distinction between commutative and noncommutative shuffle structures (Ebrahimi-Fard et al., 2014).

The pre-Lie Magnus expansion supplies closed transformations among free, Boolean, and monotone cumulants: $X$3 where $X$4, $X$5, and $X$6 denote monotone, free, and Boolean cumulant functionals, respectively. In particular, multivariate monotone cumulants admit a closed formula as sums over irreducible noncrossing partitions weighted by coefficients depending only on the associated rooted tree (Celestino et al., 2020).

Spreadability systems extend the formalism further. They generalize exchangeability systems by requiring invariance under order-preserving relabelings rather than arbitrary permutations, which brings monotone independence into the same framework. In this setting, cumulants are indexed by ordered set partitions; mixed cumulants do not generally vanish, and the correction terms are governed by Goldberg coefficients and the Campbell–Baker–Hausdorff series (Hasebe et al., 2017).

Second-order free cumulants refine the theory to fluctuations. They are defined on a second-order non-commutative probability space $X$7 using non-crossing annular partitioned permutations. For second-order free random variables $X$8 and $X$9, the low-order formulas

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

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

$$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$2

illustrate how second-order product, commutator, and anti-commutator formulas involve first- and second-order cumulant data simultaneously (George et al., 28 Jul 2025).

5. Finite free, rectangular, and algebraic generalizations

Finite free probability replaces probability measures by monic polynomials of fixed degree $$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$3. For a monic polynomial $$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$4, the finite free cumulants $$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$5 are defined through the truncated finite $$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$6-transform

$$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$7

They satisfy

$$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$8

and converge to free cumulants as $$\log \bigl(E[e^{zX}]\bigr) = \sum_{\ell=1}^n \frac{z^\ell}{\ell!}\,\kappa_\ell(X) + o(z^n) \qquad (z\to 0,\ z\in i\mathbb R).$$9. The theory thus furnishes a finite-dimensional analogue of the usual cumulant linearization of additive convolution (Arizmendi et al., 2016).

A rectangular variant adapts the same principle to the $\kappa_1(X)=E[X],$0-rectangular convolution. For a monic polynomial

$\kappa_1(X)=E[X],$1

the $\kappa_1(X)=E[X],$2-rectangular cumulants $\kappa_1(X)=E[X],$3 are defined by

$\kappa_1(X)=E[X],$4

and they linearize the rectangular finite free convolution: $\kappa_1(X)=E[X],$5 In the regime $\kappa_1(X)=E[X],$6 and $\kappa_1(X)=E[X],$7, these cumulants converge to the $\kappa_1(X)=E[X],$8-rectangular free cumulants (Cuenca, 2024).

Other generalizations modify the underlying partition lattice rather than the convolution. $\kappa_1(X)=E[X],$9-cumulants replace the full partition lattice $\kappa_2(X)=E[X^2]-E[X]^2,$0 by a smaller lattice $\kappa_2(X)=E[X^2]-E[X]^2,$1, but retain the inverse relation

$\kappa_2(X)=E[X^2]-E[X]^2,$2

under a product condition on intervals. They preserve vanishing under block independence, semi-invariance under translation, and tensorial behavior under linear maps, and they are particularly useful for tree models, hidden Markov processes, and algebraic-statistical embeddings (Zwiernik, 2010).

A different algebraic generalization treats a linear space with two commutative unital multiplications, $\kappa_2(X)=E[X^2]-E[X]^2,$3 and $\kappa_2(X)=E[X^2]-E[X]^2,$4. In that context the cumulants associated with the identity map between the two algebras satisfy

$\kappa_2(X)=E[X^2]-E[X]^2,$5

and mixed products expand as signed sums over reduced mixing forests. The construction is presented as an analogue of Leonov–Shiraev’s formula and is used to study structure constants of Jack characters (Burchardt, 2018).

Umbral calculus supplies yet another unification. Generalized Abel polynomials

$\kappa_2(X)=E[X^2]-E[X]^2,$6

generate a family of cumulants for which the classical, Boolean, and free cases are recovered by the choices $\kappa_2(X)=E[X^2]-E[X]^2,$7, $\kappa_2(X)=E[X^2]-E[X]^2,$8, and $\kappa_2(X)=E[X^2]-E[X]^2,$9, respectively. In the free case, the paper identifies the Pitman–Stanley volume polynomial as the analogue of the complete Bell polynomial in the classical moment–cumulant relation (Nardo et al., 2010).

6. Bounds, analyticity, and constructive expansions

The moment–cumulant formula yields quantitative bounds on cumulants using only moments of the same order. For a real-valued random variable $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$0 with $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$1,

$\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$2

and for $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$3 this coefficient is exactly $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$4, where $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$5 is the ordered Bell number. Using shift invariance for $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$6, one obtains the centered refinement

$\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$7

where $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$8 sums $\kappa_3(X)=E[X^3]-3E[X^2]E[X]+2E[X]^3,$9 over partitions with no singleton blocks. The asymptotic behaviors

$n$00

replace classical $n$01-type estimates by factorial-exponential coefficients dictated directly by partition combinatorics (Zhang, 7 Oct 2025).

In constructive quantum field theory and random matrix theory, cumulants also appear as analytic objects generated by source-dependent partition functions. For stable random matrix models with single-trace interaction of order $n$02, the loop vertex representation constructs matrix cumulants as absolutely convergent expansions over trees and ribbon graphs, proves analyticity in the cardioid domain

$n$03

and establishes Borel-LeRoy summability at the origin (Rivasseau, 2023).

The quartic complex matrix model admits a related but stronger variational treatment. There one distinguishes ordinary connected tensor cumulants from scalar cumulants extracted through Weingarten calculus. The resulting variational loop vertex expansion yields analyticity in a large coupling domain

$n$04

uniform-in-$n$05 factorial remainder bounds, Borel summability of the rescaled scalar cumulants, and a topological $n$06 expansion organized by genus (Rivasseau, 2 Jun 2026).

These developments show that cumulants function both as discrete combinatorial coordinates and as analytically controlled nonperturbative observables.

7. Applications in physics, statistics, and multivariate symbolic calculus

In QCD, cumulants of the topological charge distribution are defined by derivatives of the vacuum energy density in a $n$07-vacuum: $n$08 The lowest one, $n$09, is the topological susceptibility. Chiral perturbation theory computes $n$10 up to next-to-leading order, thereby generating all topological cumulants. For degenerate SU($n$11) quark masses, all cumulants depend on the same linear combination of NLO low-energy constants and the same chiral logarithm, which yields sum rules between the $n$12-flavor quark condensate and the cumulants that are free of next-to-leading order corrections (Guo et al., 2015).

In heavy-ion phenomenology, baryon-number cumulants are simultaneously event-by-event fluctuation measures and thermodynamic derivatives: $n$13 The first three cumulants are sufficient to recover the isothermal speed of sound $n$14 and its logarithmic derivative with respect to baryon density. In the regime $n$15, the paper derives the approximations

$n$16

linking fluctuation measurements to the equation of state relevant for both heavy-ion collisions and neutron stars (Sorensen et al., 2021).

In algebraic statistics, $n$17-cumulant embeddings provide polynomial coordinate systems on the probability simplex that are often better adapted than ordinary moments or classical cumulants to the combinatorics of a model. Tree cumulants simplify binary hidden tree models, and in hidden Markov processes with binary hidden states they yield especially simple parametrizations of normalized observables (Zwiernik, 2010).

A parallel symbolic development introduces cumulant polynomial sequences and their multivariable extensions. If $n$18 is the cumulant generating function of $n$19, the cumulant polynomial sequence $n$20 is defined by

$n$21

Depending on what is substituted into the indeterminate $n$22, these polynomials recover either moment sequences or cumulant sequences. Applications are given within parameter estimations, Lévy processes and random matrices, and the connection with multivariable Sheffer polynomial sequences provides a different viewpoint in characterizing exponential models (Nardo, 2016).

Across these settings, the common pattern is stable: cumulants encode connected or primitive structure, linearize the relevant convolution or composition law, and frequently reveal coordinates in which a complicated nonlinear transformation becomes additive.