Cumulants are universal homomorphisms into Hausdorff groups

Abstract

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let ProbOpen image in new window denote the convolution semigroup of all probability measures on Open image in new window with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence Open image in new window regarded as a function from ProbOpen image in new window into the additive topological group Open image in new window ofall real sequences, is universal among continuous homomorphisms from ProbOpen image in new window into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through κ. An analogous result, referring to just the first Open image in new window cumulants,holds for the semigroup Open image in new window of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from Open image in new window the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.