Dani, S. G. ; Schmidt, Klaus (2002) Affinely infinitely divisible distributions and the embedding problem Mathematical Research Letters, 9 (5). pp. 607620. ISSN 10732780

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Abstract
Let A be a locally compact abelian group and let μ be a probability measure on A. A probability measure λ on A is an affine kth root of μ if there exists a continuous automorphism ρ of A such that ρ^{k}=I (the identity transformation) and λ∗ρ(λ)∗ρ^{2}( λ)∗… ∗ρ^{k1}(λ ) = μ, and μ is affinely infinitely divisible if it has affine kth roots for all orders. Clearly every infinitely divisible probability measure is affinely infinitely divisible. In this paper we prove the converse for connected abelian Lie groups: Every affinely infinitely divisible probability measure on a connected abelian Lie group A is infinitely divisible. If G is a locally compact group, A a closed abelian subgroup of G, and μ a probability measure on G which is supported on A and infinitely divisible on G, we give sufficient conditions which ensure that μ is infinitely divisible on A.
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Deposited On:  16 Dec 2011 09:25 
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