Binomial Type - Cumulants and Moments

Cumulants and Moments

The sequence κ_n of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the cumulants of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled cumulant. Thus

the nth cumulant

and

the nth moment.

These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.

Let

be the (formal) cumulant-generating function. Then

is the delta operator associated with the polynomial sequence, i.e., we have