Combinant

Template:Short description Template:One source 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