Probability theory and stochastic processes Definition

Cumulant generating function

Builds on Moment-generating function Moments are derivatives of the moment-generating function at zero Moment-generating function of a sum of independent random variables

Definition

Let $X$ be a real random variable with moment-generating function $M_X$ . The cumulant generating function of $X$ is $K_X(t) = \ln M_X(t) = \ln E\left[e^{tX}\right],$ for each $t \in \mathbb{R}$ with $M_X(t) < \infty$ . If $M_X$ is finite on an open interval around $0$ , the cumulants $\kappa_n$ of $X$ are the coefficients in $K_X(t) = \sum_{n=1}^{\infty} \kappa_n t^n / n!$ , so $\kappa_n = K_X^{(n)}(0)$ .

Remarks

The first cumulants are $\kappa_1 = E[X]$ , $\kappa_2 = \operatorname{Var}(X)$ and $\kappa_3 = E[(X - E[X])^3]$ . For independent $X$ and $Y$ , $K_{X+Y} = K_X + K_Y$ by the product rule for moment-generating functions. So cumulants of independent sums add. For $X \sim \mathcal{N}(\mu, \sigma^2)$ , $K_X(t) = \mu t + \sigma^2 t^2 / 2$ . For $X \sim \operatorname{Poisson}(\lambda)$ , $K_X(t) = \lambda(e^t - 1)$ , so all cumulants equal $\lambda$ . $K_X$ is convex on the interval where it is finite.

Reading path

To understand this page from the ground up, read these 64 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Random variable with a density definition
Joint probability distribution definition
Probability mass function definition
Mutual independence of events definition
Expected value definition
Independence of two random variables definition
Probability density function definition
Marginal distribution definition
Independent sigma-algebras definition
Law of the unconscious statistician statement
Independence of random variables definition
Independence means the joint law is the product of the marginals statement
Moment definition
Moment-generating function definition
Expectation of a product of independent random variables statement
Moment-generating function of a sum of independent random variables statement
Moments are derivatives of the moment-generating function at zero statement
Cumulant generating function definition

Tag	`0Q2QC81VTW`
URI	`m2://0Q2QC81VTW`
Kind	definition
Writing	W0 stub
MSC	60E10
Defines	Cumulant generating function
Uses	Moment-generating function, Moments are derivatives of the moment-generating function at zero, Moment-generating function of a sum of independent random variables

Cumulant generating function

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here