What are cumulants?

Abstract

Let $\mathcal{P}$ be the set of all probability measures on $\mathbb{R}$ possessing moments of every order. Consider $\mathcal{P}$ as a semigroup with respect to convolution. After topologizing $\mathcal{P}$ in a natural way, we determine all continuous homomorphisms of $\mathcal{P}$ into the unit circle and, as a corollary, those into the real line. The latter are precisely the finite linear combinations of cumulants, and from these all the former are obtained via multiplication by $i$ and exponentiation.