# 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 $X$.

Title cumulant generating function CumulantGeneratingFunction 2013-03-22 16:16:24 2013-03-22 16:16:24 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 17 Andrea Ambrosio (7332) Definition msc 60E05 MomentGeneratingFunction CharacteristicFunction2