Partly divisible probability measures on locally compact Abelian groups

Details Partly divisible probability measures on locally compact Abelian groups Partly divisible probability measures on locally compact Abelian groups

2005-01-12

arxiv.org/abs/math/0501185v1

Mathematische Nachrichten 213 (2000), 5-15

Mathematics

Probability

15 pages

Scientific paper

A notion of admissible probability measures $\mu$ on a locally compact Abelian group (LCA-group) $G$ with connected dual group $\hat G=\R^d\times \T^n$ is defined. To such a measure $\mu$, a closed semigroup $\Lambda(\mu)\subseteq (0,\infty)$ can be associated, such that, for $t\in \Lambda(\mu)$, the Fourier transform to the power $t$, $(\hat \mu)^t$, 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 $\Z_2$, in the case of LCA-groups with non connected dual group, there is no canonical definition of the set $\Lambda(\mu)$.

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