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 $K'(0)=\lambda$ $λ$ .

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