Combinant

In probability theory, the combinants of a non-negative integer-valued random variable are coefficients in the power-series expansion of the logarithm of its probability-generating function. They have been used particularly in the analysis of particle multiplicity distributions.

Definition

Let $X$ be a non-negative integer-valued random variable with probability mass function

$p_{k}=\Pr(X=k),\qquad k=0,1,2,\ldots ,$

and suppose that $p_{0}>0$. Its probability-generating function is

$G_{X}(z)=\operatorname {E} [z^{X}]=\sum _{k=0}^{\infty }p_{k}z^{k}.$

The combinants $c_{n}$ are defined by the expansion

$\log G_{X}(z)=\log p_{0}+\sum _{n=1}^{\infty }c_{n}z^{n}.$

Equivalently,

$c_{n}={\frac {1}{n!}}\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} z^{n}}}\log G_{X}(z)\right|_{z=0},\qquad n\geq 1.$

This definition requires a nonzero probability of $X=0$, so that the logarithm of the generating function is defined in a neighborhood of the origin.

If

$M_{X}(s)=\operatorname {E} [e^{sX}]$

is the moment-generating function of $X$, then, wherever the expressions exist,

$G_{X}(z)=M_{X}(\log z).$

Defining the shifted generating function

$H_{X}(t)=G_{X}(1+t)=M_{X}!\left(\log(1+t)\right)=\operatorname {E} [(1+t)^{X}],$

gives the equivalent expression

$c_{n}={\frac {1}{n!}}\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}\log H_{X}(t)\right|_{t=-1}.$

Properties

Like cumulants, combinants are additive for sums of independent random variables. If $X$ and $Y$ are independent, their probability-generating functions satisfy

$G_{X+Y}(z)=G_{X}(z)G_{Y}(z),$

and therefore

$c_{n}(X+Y)=c_{n}(X)+c_{n}(Y).$

For a Poisson distribution with mean $\lambda$,

$G_{X}(z)=\exp(\lambda (z-1)),$

so that

$c_{1}=\lambda$

and $c_{n}=0$ for $n\geq 2$.

Combinants are related to, but distinct from, factorial cumulants. Combinants are obtained from derivatives of $\log G_{X}(z)$ at $z=0$, whereas factorial cumulants are obtained from derivatives at $z=1$.

For non-negative integer-valued infinitely divisible distributions, the probability-generating function has a compound-Poisson form, and the corresponding combinants are non-negative. Consequently, the occurrence of a negative combinant is incompatible with infinite divisibility.

See also

• Compound Poisson distribution
• Cumulant
• Factorial moment
• Moment-generating function
• Probability-generating function

Further reading

• Kittel, W.; De Wolf, E. A. (2005). Soft Multihadron Dynamics. World Scientific. pp. 306–307. ISBN 978-981-256-295-1.