Affinely infinitely divisible distributions and the embedding problem

Dani, S. G. ; Schmidt, Klaus (2002) Affinely infinitely divisible distributions and the embedding problem Mathematical Research Letters, 9 (5). pp. 607-620. ISSN 1073-2780

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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 k-th root of μ if there exists a continuous automorphism ρ of A such that ρk=I (the identity transformation) and λ∗ρ(λ)∗ρ2( λ)∗… ∗ρk-1(λ ) = μ, and μ is affinely infinitely divisible if it has affine k-th 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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Source:Copyright of this article belongs to International Press.
ID Code:74513
Deposited On:16 Dec 2011 09:25
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