Combinant

In the mathematical theory of probability, the combinants c_n of a random variable X are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as

$G_X(t)=M_X(\log (1+t))$

which can be expressed directly in terms of a random variable X as

$G_X(t):=E[(1+t{)}^X],t\in ℝ,$

wherever this expectation exists.

The nth combinant can be obtained as the nth derivatives of the logarithm of combinant generating function evaluated at –1 divided by n factorial:

$c_n=\frac{1}{n!}\frac{{\partial }^n}{\partial t^n}\log (G(t)){|}_{t=-1}$

Important features in common with the cumulants are:

• the combinants share the additivity property of the cumulants;
• for infinite divisibility (probability) distributions, both sets of moments are strictly positive.

References

• Google Books