Cumulants and moments (Q797123)

In [Lett. Math. Phys. 2, 207-217 (1978; Zbl 0379.16018)] the concept of quasi-multiplication on rings and algebras was introduced. In the present paper the author looks at a \({}^*\)-algebra of terminating sequences whose entries are complex numbers. The set of normalized linear functions on this algebra is then equipped with a quasi-multiplication and also with a multiplication. The two operations are intimately related, but both have their own right. This is shown for sequences of moments (positive linear functionals, states) and their corresponding sequences of cumulants. Necessary and sufficient conditions for a sequence to be a sequence of moments, and a sequence to be a sequence of cumulants, are also given. Further studies, not mentioned in this paper, include higher derivatives of composite functions (Faà di Bruno), the construction of new sequences of moments from given sequences of moments, and the statement, that every sequence of moments is a sequence of cumulants.