What are cumulants? (Q1805572)

Summary: 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. 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 Halász. 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 \textit{I. Z. Ruzsa} and \textit{G. J. Székely} [Monatsh. Math. 95, 235-239 (1983; Zbl 0509.43004), Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 263-269 (1985; Zbl 0554.60026) and ``Algebraic probability theory'' (1988; Zbl 0653.60012)].