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Cumulants are universal homomorphisms into Hausdorff groups
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  • Published: 25 May 2004

Cumulants are universal homomorphisms into Hausdorff groups

  • Lutz Mattner1 

Probability Theory and Related Fields volume 130, pages 151–166 (2004)Cite this article

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Abstract.

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let Prob denote the convolution semigroup of all probability measures on with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence regarded as a function from Prob into the additive topological group ofall real sequences, is universal among continuous homomorphisms from Prob into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through κ. An analogous result, referring to just the first cumulants,holds for the semigroup of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.

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Authors and Affiliations

  1. Universität zu Lübeck, Institut für Mathematik, Wallstraße 40, 23560, Lübeck, Germany

    Lutz Mattner

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  1. Lutz Mattner
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Correspondence to Lutz Mattner.

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Mathematics Subject Classification (2000): 60B15, 60E10, 60G50

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Mattner, L. Cumulants are universal homomorphisms into Hausdorff groups. Probab. Theory Relat. Fields 130, 151–166 (2004). https://doi.org/10.1007/s00440-004-0354-y

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  • Received: 12 November 2003

  • Revised: 14 February 2004

  • Published: 25 May 2004

  • Issue Date: October 2004

  • DOI: https://doi.org/10.1007/s00440-004-0354-y

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Key words and phrases:

  • Algebraic probability theory
  • Characteristic functions
  • Quotients of positive definite functions
  • Sums of independent random variables
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