What Are Cumulants ?

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.

We obtain as corollaries similar results for the probability measures with some or no moments finite, and characterizations of constant multiples of cumulants as affinely equivariant and convolution-additive functionals. The ``no moments''-case yields a theorem of Hal\'asz. Otherwise our results appear to be new even when specialized to yield characterizations of the expectation or the variance.

Our basic tool is a refinement of the convolution quotient representation theorem for signed measures of Ruzsa \& Székely.

1991 Mathematics Subject Classification: 60E05, 60E10, 60-03.

Keywords and Phrases: Additive functional, characteristic function, character, convolution, equivariance, expectation, Hal\'asz, historical note, homomorphism, mean, moment, multiplicative functional, Ruzsa, semi-invariant, semiinvariant, semigroup, Sz\'ekely, variance.

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