Addendum to the paper 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 ρ k = I (the identity transformation) and λ * ρ(λ) * ρ 2 (λ) * • • • * ρ k−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.