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
Canonical name	CumulantGeneratingFunction
Date of creation	2013-03-22 16:16:24
Last modified on	2013-03-22 16:16:24
Owner	Andrea Ambrosio (7332)
Last modified by	Andrea Ambrosio (7332)
Numerical id	17
Author	Andrea Ambrosio (7332)
Entry type	Definition
Classification	msc 60E05
Related topic	MomentGeneratingFunction
Related topic	CharacteristicFunction2