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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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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DOI: https://doi.org/10.1007/s00440-004-0354-y
Key words and phrases:
- Algebraic probability theory
- Characteristic functions
- Quotients of positive definite functions
- Sums of independent random variables