On Expressing Moments in Terms of Cumulants and Vice Versa

Abstract

The procedure for generating moments in terms of cumulants commonly discussed in texts is that of equating coefficients in the power series expansions of the defining relationship between the respective generating functions, viz. M(t) = exp [C(t)]. If this procedure is followed, the task of identifying and collecting coefficients soon becomes cumbersome and tedious, especially in the multivariate case. The recursive techniques discussed in this paper are simple to apply and quite general-and not widely known. They allow one to express moments (multivariate as well as univariate) in terms of cumulants, and vice versa, with little more effort than it takes to write the result. Any path consisting of single forward steps may be followed, and no known starting point is required (aside from the knowledge that the zeroth moment is unity).