# Cumulant-generating function

The cumulant-generating function (CGF) of a random variable X is the natural logarithm of its moment-generating function, K(t) = log E[e^{tX}] = log M(t), and its derivatives at zero, the cumulants, encode a distribution's shape in a form that adds exactly under convolution of independent variables.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[2](https://www.statlect.com/fundamentals-of-probability/cumulant-generating-function)</sup> The logarithm is the whole point of the transform: because the moment-generating function of a sum of independent random variables is a product, taking logs turns that product into a sum, so each cumulant of the sum is simply the sum of the corresponding cumulants.<sup>[3](https://www.sfu.ca/~lockhart/richard/830/20_3/lectures/generating_functions/notes.pdf)</sup><sup> • </sup><sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> [Raw moments](https://www.edgepedia.org/moment-mathematics) of a sum satisfy no such clean rule; variance is the famous exception among them precisely because it is a cumulant.

| Key fact | Statement |
|---|---|
| Definition | \(K(t) = \log E[e^{tX}] = \log M(t)\); cumulants are the Taylor coefficients: \(K(\xi) = \sum \kappa_r \xi^r / r!\), \(\kappa_0 = 0\)<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |
| Extraction | \(\kappa_{n} = K^{(n)}(0)\) from the MGF, or \(\kappa_{n} = K^{(n)}(0)/i^{n}\) from the log characteristic function<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> |
| Additivity | For independent X, Y: \(\kappa_{n}(X + Y) = \kappa_{n}(X) + \kappa_{n}(Y)\) at every order n<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> |
| Translation/scaling | For \(r \geq 2\), \(\kappa_r(a + bX) = b^{r} \cdot \kappa_r\), independent of \(a\); \(\kappa_1(a + bX) = a + b \cdot \kappa_1\)<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |
| Normal | \(N(\mu, \sigma^2)\) has CGF \(\xi \mu + \xi^2 \sigma^2 / 2\), so \(\kappa_n = 0\) for \(n \geq 3\)<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |
| Poisson | CGF \(\mu(e^{\xi} - 1)\); every cumulant equals the mean μ<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |
| Existence failure | The lognormal has finite moments of all orders but its CGF diverges for every ξ > 0<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |

## Definition and basic form

For a random variable \(X\) with moment-generating function \(M(t) = E[e^{tX}]\), the cumulant-generating function is \(K(t) = \log M(t)\). Expanding \(K\) about the origin gives \(K(\xi) = \log M(\xi) = \Sigma_r \kappa_r \xi^r / r!\), and the coefficients \(\kappa_r\) are the cumulants, with \(\kappa_0 = 0\).<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup>

The logarithm restructures what the transform records. In M(t), the information about sums of independent variables appears as products; in K(t) it appears as sums. Richard Lockhart's Simon Fraser University lecture notes put the motivation directly: since the logarithm of a product is a sum of logarithms, taking logs of the moment-generating function yields cumulants "which add up properly."<sup>[3](https://www.sfu.ca/~lockhart/richard/830/20_3/lectures/generating_functions/notes.pdf)</sup> The same log-transform applies to the other generating functions: log M (of the [MGF](https://www.edgepedia.org/moment-generating-function)), log G (of the probability-generating function for integer-valued variables), and K = log φ (of the characteristic function) are each called cumulant-generating functions, and each generates the same cumulants.<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup>

## Cumulants as derivatives at zero

Cumulants are read off by differentiation: \(\kappa_{n} = K^{(n)}(0)\) when \(K\) comes from the MGF. When \(K\) is built from the characteristic function \(\varphi\), the \(i\)-th cumulant is \(\kappa_{n} = K^{(n)}(0)/i^{n}\).<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> In software terms, the [Wolfram Language](https://www.edgepedia.org/wolfram-language) defines the CGF as Log of the moment-generating function and extracts the \(i\)-th cumulant as SeriesCoefficient[cgf, {t, 0, i}] i!.<sup>[5](https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html)</sup>

The first cumulants, in terms of raw moments \(\mu_{r} = E[X^{r}]\), are:<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup>

- κ1 = μ1 (the mean)
- κ2 = μ2 − μ1² (the variance σ²)<sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup>
- κ3 = μ3 − 3μ2μ1 + 2μ1³ (the third central moment)
- κ4 = μ4 − 4μ3μ1 − 3μ2² + 12μ2μ1² − 6μ1⁴

A worked example from P. McCullagh's STAT 306 notes illustrates the extraction: a distribution whose expansion yields μ2 = κ2 = 1/3 and μ4 = 1 has κ4 = 2/3, since κ4 = μ4 − 3μ2² = 1 − 3(1/9).<sup>[7](https://www.stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf)</sup>

## Key properties: additivity, scaling, translation

**Additivity.** If \(S = X + Y\) with \(X\) and \(Y\) independent, then \(M_{S}(\xi) = M_{X}(\xi) \cdot M_{Y}(\xi)\) and therefore \(K_{S}(\xi) = K_{X}(\xi) + K_{Y}(\xi)\); differentiating \(n\) times at zero gives the theorem Wichura calls cumulants accumulate: \(\kappa_{n}(S) = \kappa_{n}(X) + \kappa_{n}(Y)\) for every \(n\).<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> The same source notes this relation is much simpler than the corresponding relation for moments, which is the reason for the name "cumulants." For a sum of independent \(X_{i}\), log of the MGF of \(Y\) is the sum of the logs, so \(\kappa_{r}(Y) = \sum \kappa_{r}(X_{i})\).<sup>[8](https://www.sfu.ca/~lockhart/richard/830/11_3/lectures/generating_functions/web.pdf)</sup>

Independence is essential, not decorative. Without it, cross-terms reappear and \(\kappa_{n}(X+Y) \neq \kappa_{n}(X) + \kappa_{n}(Y)\) in general, exactly as raw moments fail to add; applying the rule to correlated quantities such as overlapping asset returns is a common error.<sup>[9](https://quantmemo.com/concepts/cumulant-generating-functions)</sup> A related slip is averaging standardized skewness or kurtosis numbers across positions instead of combining the underlying cumulants first and standardizing only the final sum.<sup>[9](https://quantmemo.com/concepts/cumulant-generating-functions)</sup>

**Translation and scaling.** The theorem of semi-invariance states \(\kappa_n(X + b) = \kappa_n(X) + b\) if \(n = 1\), and \(\kappa_n(X)\) if \(n > 1\).<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> Combined with scaling: for \(r \geq 2\), the \(r\)-th cumulant of the affine transformation \(a + bX\) is \(b^{r} \cdot \kappa_r\), independent of \(a\).<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[7](https://www.stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf)</sup> Shifting a distribution by a constant changes its location but not its shape, and all cumulants beyond the first measure shape, not location; that is why Thiele, who first defined them, called them semi-invariants.<sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup>

## By the numbers: CGFs of standard distributions

| Distribution | CGF K(ξ) | Cumulants |
|---|---|---|
| Normal \(N(\mu, \sigma^2)\) | \(\xi \mu + \xi^2 \sigma^2 / 2\) | \(\kappa_1 = \mu\), \(\kappa_2 = \sigma^2\), \(\kappa_n = 0\) for \(n \geq 3\)<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup> |
| Poisson(μ) | μ(e^ξ − 1) | all cumulants equal μ<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |
| Exponential(mean λ) | −log(1 − λξ) | \(\kappa_{r} = \lambda^{r} \cdot (r-1)!\)<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup> |
| Bernoulli(π) | \(\log(1 - \pi + \pi e^{\xi})\) | cumulants from derivatives at zero<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> |

Two rows repay attention. The exponential row, \(\kappa_{r} = (n-1)! \cdot \lambda^{n}\) for the mean-λ version,<sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup> and the Poisson row, in which all cumulants equal the mean.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup>

## Gaussianity, the CLT, and the Marcinkiewicz theorem

The normal distribution has the simplest cumulant list: K1 = μ, K2 = σ², and Kn = 0 for n ≥ 3.<sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup> This is not an accident of one distribution. Marcinkiewicz (1939) showed that the normal is the only distribution whose cumulant-generating function is a polynomial, that is, the only distribution having a finite number of non-zero cumulants.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> In cumulant language, "Gaussian" means exactly that all cumulants of order three and above vanish; any polynomial CGF other than a quadratic is impossible.

Additivity then makes the central limit theorem a scaling computation. For \(n\) i.i.d. summands, the \(r\)-th cumulant of the standardized sum scales as \(n^{1-r/2} \cdot \kappa_r\), which tends to zero for \(r \ge 3\): variance (\(r = 2\)) survives standardization at level 1, third cumulants shrink like \(n^{-1/2}\), fourth like \(n^{-1}\), and so on.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> The same arithmetic explains why variance adds across independent risks (\(\kappa_2\) adds at every order) and why correct combination of skewness and kurtosis across independent components means combining their cumulants \(\kappa_3\) and \(\kappa_4\), then standardizing the totals.<sup>[9](https://quantmemo.com/concepts/cumulant-generating-functions)</sup>

## When the CGF fails to exist: heavy tails and the characteristic function

Finiteness of all moments does not guarantee a CGF. The lognormal distribution is the standard counterexample: it has finite moments \(\mu_{r} = e^{r^{2}/2}\) of all orders, but its MGF is finite for \(t \leq 0\) and infinite for \(t > 0\), so its CGF is not finite in any open neighborhood of zero.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> For such distributions the MGF route is closed entirely, even though every moment exists.

The characteristic-function route remains partially open. Since \(\kappa_n = K^{(n)}(0)/i^n\) for \(K = \log \varphi\),<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> cumulants can be defined through finite order whenever \(\log \varphi\) is differentiable that many times at zero. For the [Cauchy distribution](https://www.edgepedia.org/cauchy-distribution), the log of the characteristic function is not differentiable at zero, so even its first cumulant is not defined by this route; Student's t with five degrees of freedom has moments, and hence cumulants, through order four.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> More generally, formula (5) of Wichura's notes is used to define the first n cumulants when X has only an n-th moment, i.e. when \(\mathrm{E}(|X|^n) < \infty\).<sup>[4](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)</sup> Long-tailed distributions can therefore have a finite prefix of cumulants and nothing beyond: for some distributions with long tails, such as a Pareto-type law, cumulants of only several low orders are defined.<sup>[5](https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html)</sup>

Multivariate cumulants extend the same idea: coefficients κ^{r,s}, κ^{r,s,t}, ... in the Taylor expansion of log M(ξ) (written in [Einstein summation notation](https://www.edgepedia.org/einstein-notation)) serve as joint cumulants.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup> How mixed cumulants behave under independence diagnostics is not covered in the sources retained here.

## History

The notion of a cumulant traces to the Danish scientist T. N. Thiele, who derived them in 1889 and called them semi-invariants; Hald (2000) credits Thiele with the first derivation.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[6](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)</sup> The modern theory of cumulants and the associated k-statistics begins with Fisher's 1929 paper, in which Fisher called the quantities \(\kappa_{j}\) cumulative moment functions.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[7](https://www.stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf)</sup> The term "cumulant" itself was suggested by Hotelling (1933) in a letter to Fisher, who approved of the coinage; Lauritzen (2002) presents an overview, translation, and reprinting of much of this early work.<sup>[1](http://www.scholarpedia.org/article/Cumulants)</sup><sup> • </sup><sup>[7](https://www.stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf)</sup> The name records the defining property: cumulants accumulate over independent sums.

## References

1. [Cumulants - Scholarpedia](http://www.scholarpedia.org/article/Cumulants)
2. [Cumulant generating function | Formula, derivatives, proofs - StatLect](https://www.statlect.com/fundamentals-of-probability/cumulant-generating-function)
3. [Moment Generating Functions (SFU STAT 830 notes, Lockhart)](https://www.sfu.ca/~lockhart/richard/830/20_3/lectures/generating_functions/notes.pdf)
4. [Cumulants (Wichura, Stat 304, University of Chicago)](https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf)
5. [CumulantGeneratingFunction — Wolfram Documentation](https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html)
6. [On the Combinatorics of Cumulants (Rota & Shen, 2000)](http://oerior.uniud.it/wp-content/uploads/2018/12/RotaShen2000.pdf)
7. [Cumulants (P. McCullagh, STAT 306, University of Chicago)](https://www.stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf)
8. [STAT 830 Generating Functions (SFU)](https://www.sfu.ca/~lockhart/richard/830/11_3/lectures/generating_functions/web.pdf)
9. [Cumulants and the Cumulant Generating Function, Explained | Quant Memo](https://quantmemo.com/concepts/cumulant-generating-functions)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability distributions › Characteristic and generating functions › Cumulants and cumulant generating functions*

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