ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60 Mattner, Lutz, Universität Hamburg, Germany What are cumulants? We characterize cumulants [= expectation, variance, …] by their useful properties: Let n ∈ {0, 1, …} and let P_{n} be the set of all probability measures P on ℝ with ∫ |x|^{n} dP (x) < ∞. Consider P_{n} as a semigroup with respect to convolution. Topologize P_{n} by the weighted total variation distance (P, Q) ↦ ∫ (1 + |x|)^{n} d|P − Q|(x). Theorem The continuous homomorphisms from P_{n} into the additive group ℝ are precisely the linear combinations of cumulants of order ≤ n. Corollary The scale equivariant ones are the constant multiples of cumulants. For n = 0, the Theorem yields a result of Halász [given in Rusza & Székely (1988), Algebraic Probability Theory, Wiley], namely: The only continuous homomorphism on P_{0} is 0.