## What are cumulants?

### Summary

Summary: Let $\cP$ be the set of all probability measures on $\R$ possessing moments of every order. Consider $\cP$ as a semigroup with respect to convolution. After topologizing $\cP$ in a natural way, we determine all continuous homomorphisms of $\cP$ 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.

### Mathematics Subject Classification

60E05, 60E10, 60-03

### Keywords/Phrases

additive functional, characteristic function, character, convolution, equivariance, expectation, halász, historical note, homomorphism, mean, moment, multiplicative functional, ruzsa, semi-invariant, semiinvariant, semigroup, székely, variance