Partly divisible probability measures on locally compact Abelian groups

2005/01/12 by S. Albeverio, H. Gottschalk, J. -L. Wu
Mathematics · #math.PR #msc:60B15 #msc:60E10

paper · pdf

published as Mathematische Nachrichten 213 (2000), 5-15 · 15 pages

arxiv created 2005/01/12 · arxiv updated 2009/12/01

Abstract

A notion of admissible probability measures μ on a locally compact Abelian group (LCA-group) G with connected dual group G=\R^d× \Tⁿ is defined. To such a measure μ, a closed semigroup Λ(μ)⊆ (0,∞) can be associated, such that, for t∈ Λ(μ), the Fourier transform to the power t, ( μ)^t, is a characteristic function. We prove that the existence of roots for non admissible probability measures underlies some restrictions, which do not hold in the admissible case. As we show for the example \Z₂, in the case of LCA-groups with non connected dual group, there is no canonical definition of the set Λ(μ).

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