Combinant

{{Short description|Mathematical theory}}

{{one source |date=March 2024}}

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

:$G\_X(t)=M\_X(\backslash log(1+t))$

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

:$G\_X(t)\; :=\; E\backslash left[(1+t)^X\backslash right],\; \backslash quad\; t\; \backslash in\; \backslash mathbb\{R\},$

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:

:$c\_n\; =\; \backslash frac\{1\}\{n!\}\; \backslash frac\{\backslash partial\; ^n\}\{\backslash partial\; t^n\}\; \backslash log(G\; (t))\; \backslash bigg|\_\{t=-1\}$

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.