Combinant

In the mathematical theory of probability, the combinants cn 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 ) ) {\displaystyle 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 , {\displaystyle G_{X}(t):=E\left[(1+t)^{X}\right],\quad t\in \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 = 1 n ! ∂ n ∂ t n log ⁡ ( G ( t ) ) | t = − 1 {\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\log(G(t)){\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.

Mathematical theory

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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 ) ) {\displaystyle 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 , {\displaystyle G_{X}(t):=E\left[(1+t)^{X}\right],\quad t\in \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 = 1 n ! ∂ n ∂ t n log ⁡ ( G ( t ) ) | t = − 1 {\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\log(G(t)){\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.

References

[edit]

• Kittel, W.; De Wolf, E. A. Soft Multihadron Dynamics. pp. 306 ff. ISBN 978-9812562951. Google Books

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