Affinely infinitely divisible distributions and the embedding problem

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 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.