Combinant - Alchetron, The Free Social Encyclopedia

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 ∈ 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 = 1 n ! ∂ n ∂ 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

Combinant Wikipedia

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