# cumulant generating function

Given a random variable^{} $X$, the *cumulant generating function* of $X$ is the following function:

$${H}_{X}(t)=\mathrm{ln}E[{e}^{tX}]$$ |

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 |