Affinely infinitely divisible distributions and the embedding problem

Abstract

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.

On the embedding problem for infinitely divisible distributions on certain Lie groups with toral center

“…In this section we deduce the corollary stated in the Introduction and describe the connection with the work in [4].…”

“…The question of embeddability of infinitely divisible measures on the groups as above that are supported on A (and hence in N ) arose in the course of the work presented in [4]. Using the affirmative answer to the question the following result is obtained in [4] (strengthening the result in an earlier version of that paper); we recall here only a special case, as an illustration of application of Theorem 1.1, and refer the reader to [4] for other related results, including a stronger theorem for the case when A is a vector space, and a technical result in which G can be a general locally compact group.…”

“…More generally one can deduce the following in this respect (see § 5 for proof). The question of embeddability of measures as in Theorem 1.1, for a certain subclass of the groups, arose in the course of the work presented in [4]. In § 5 we describe the class of examples involved there, and the application made in [4] of the result.…”

On the embeddability of certain infinitely divisible probability measures on Lie groups

“…For any closed subset S of G we denote by P (S) the subspace of P (G) consisting of the probability measures whose support is contained in S. For a subset Λ of probability measures on G we denote by G(Λ) the smallest closed subgroup of G containing the supports of all λ ∈ Λ, and by Z(Λ) the centraliser of G(Λ) in G. For λ ∈ P (G), G({λ}) and Z({λ}) will be written as G(λ) and Z(λ) respectively. For λ ∈ P (G) we denote by N(λ) the normaliser of G(λ) in G. We recall that given a (convolution) root ρ of λ, there exists x ∈ N(λ) such that ρ ∈ P (xG(λ)); furthermore, if ρ n = λ then for every such x we have x n ∈ G(λ) (see, for instance, [11], Lemma 2.2).…”

On the embedding problem for infinitely divisible distributions on certain Lie groups with toral center

“…The weaker version of Theorem 1.1 stated above would not suffice where the original statement is used in [2], but it turns out that the proof of the result in [2] can be completed without recourse to Theorem 1.1 of [1]; see [3].…”