Cumulant Generating Function¶

Definition¶

If the moment generating function (MGF) $M_X(t)=\mathbb{E}[\mathrm{e}^{tX}]$ is finite in a neighbourhood of zero, the cumulant generating function is $K_X(t)=\log M_X(t)$ on that domain. Its derivatives at zero are called cumulants.

Interpretation¶

Taking logarithms turns products of independent MGFs into sums. This makes cumulants a convenient way to describe aggregate risk.

Graph of the Poisson cumulant generating functions lambda(e^t - 1) for lambda = 1, 2 and 3: the curve for X + Y is the pointwise sum of the curves for X and Y, all through the origin.

The figure plots the Example's $K(t)=\lambda(\mathrm{e}^t-1)$ for independent $X\sim\operatorname{Pois}(1)$ and $Y\sim\operatorname{Pois}(2)$ (rates chosen for the figure) and for $X+Y\sim\operatorname{Pois}(3)$ $Pois (3)$ . At every $t$ the green curve is the sum of the blue and red ones, which is the additivity $K_{X+Y}=K_X+K_Y$ $K_{Y}$ . All three pass through the origin because $K(0)=\log1=0$ $0$ , and their slopes there are $1$ , $2$ and $3$ , the means $K'(0)=\lambda$ $λ$ .

Change the two Poisson means and move $t$ : $K_{X+Y}(t)=K_X(t)+K_Y(t)$ at every $t$ , so $X+Y$ is Poisson with mean $\lambda_X+\lambda_Y$ $λ_{Y}$ , and the first two derivatives at 0 both equal the mean.

The MGF of an independent sum is the product of the component MGFs. Taking its logarithm converts that product into addition, which explains the additive cumulant formula.

Key Results¶

$K_X'(0)=\mathbb{E}[X]$ and $K_X''(0)=\operatorname{Var}(X)$ $Var (X)$ .
For independent $X,Y$ with suitable MGFs, $K_{X+Y}=K_X+K_Y$ $K_{Y}$ .
$K_{aX+b}(t)=bt+K_X(at)$ for constants $a,b$ on the common domain.
Existence of ordinary moments does not ensure a finite MGF for positive arguments.

Examples¶

For $X\sim\operatorname{Pois}(\lambda)$ $Pois (λ)$ , $K_X(t)=\lambda(\mathrm{e}^t-1)$ $1)$ . Its first two derivatives at zero both equal $\lambda$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 1

Aliases: CGF, log moment generating function, cumulant-generating function, log-MGF Keywords: K_X(t) = log M_X(t), cumulants, K'(0) = E[X], K''(0) = Var(X), additivity for independent sums, Poisson CGF λ(e^t - 1) Symbols: lambda

Last updated: 17 October 2026 21:53