Lutz Mattner Cumulants are universal homomorphisms into Hausdorff groups

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let Prob∞(R) denote the convolution semigroup of all probability measures on R with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence κ = (κ : ∈ N), regarded as a function from Prob∞(R) into the additive topological group R∞ of all real sequences, is universal among continuous homomorphisms from Prob∞(R) 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 r ∈ N0 cumulants, holds for the semigroup Probr (R) of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from Prob(R), the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.