cumulant generating function

Given a random variable $X$, the cumulant generating function of $X$ is the following function:

$$H_X(t)=\ln E[e^{tX}]$$

for all $t\in R$ in which the expectation converges.

In other , the cumulant generating function is just the logarithm of the moment generating function.

The cumulant generating function of $X$ is defined on a (possibly degenerate) interval containing $t=0$; one has $H_X(0)=0$; moreover, $H_X(t)$ is a convex function (http://planetmath.org/ConvexFunction). (Indeed, the moment generating function is defined on a possibly degenerate interval containing $t=0$, which image is a positive interval containing $t=1$; so the logarithm is defined on the same interval on which is defined the moment generating function.)

The $k$th-derivative of the cumulant generating function evaluated at zero is the $k$th cumulant of $X$.