cumulant generating function

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


for all tR 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 HX(0)=0; moreover, HX(t) is a convex function ( (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 kth-derivative of the cumulant generating function evaluated at zero is the kth 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