An introduction to the embedding problem for probabilities on locally compact groups

McCrudden, M.

doi:10.1515/9783110811186.147

Cited by 9 publications

(4 citation statements)

References 27 publications

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“…A probability measure µ on a locally compact group is said to be embeddable if there exists a continuous one-parameter convolution semigroup {µ t } t≥0 such that µ 1 = µ. The question of embeddability of infinitely divisible probability measures, the so-called 'embedding problem', has been studied in literature to a considerable extent (see [4] and [8] for some details and other references). In this respect we note that since every infinitely divisible probability measure on any compactly generated abelian Lie group is embeddable (see [7], § 3.5.11), Corollary 1.2 implies the following.…”

Section: Then µ Is Infinitely Divisible On Amentioning

confidence: 99%

Addendum to the paper Affinely infinitely divisible distributions and the embedding problem

Dani

Schmidt

2005

Mathematical Research Letters

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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 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.

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Section: Then µ Is Infinitely Divisible On Amentioning

confidence: 99%

Addendum to the paper Affinely infinitely divisible distributions and the embedding problem

Dani

Schmidt

2005

Mathematical Research Letters

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“…The question of embeddability of infinitely divisible probability measures, the so-called 'embedding problem', has been studied in literature to a considerable extent (see [4] and [8] for some details and other references). In this respect we note that since every infinitely divisible probability measure on any compactly generated abelian Lie group is embeddable (see [7], § 3.5.11), Corollary 1.2 implies the following.…”

Section: Then µ Is Infinitely Divisible On Amentioning

confidence: 99%

Affinely infinitely divisible distributions and the embedding problem

Dani

Schmidt

2002

Mathematical Research Letters

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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 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.

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“…The embedding problem, originated by Parthasarathy [21] (for compact groups, see also [23]), is known as the problem of characterizing the locally compact groups admitting the embedding property. We refer to Chapter III of [13] and the survey articles of Heyer [14,15] and McCrudden [17,18] for an overview of the (recent) developments and open problems concerning the embedding problem. In fact, the p-adic solenoid G = S p is known as an example of a locally compact group not having the embedding property, since any Dirac measure μ = δ y with y ∈ S p \S arc p is not continuously embeddable.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the p-adic solenoid G = S p is known as an example of a locally compact group not having the embedding property, since any Dirac measure μ = δ y with y ∈ S p \S arc p is not continuously embeddable. In this sense, as an example of Dixmier [8], the p-adic solenoid is what is called indecent (in Definition 3.5 of [18]) to the embedding property. But every Dirac measure is rationally embeddable by Satz 11 of Böge [6], since the p-adic solenoid as a compact group is strongly root compact (see Definition 3.1 of [18]) by Theorem 3.10 together with Example 3.11 of [18].…”

Section: Introductionmentioning

confidence: 99%

Explicit representation of roots on p-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

Becker-Kern

2007

Proc Math Sci

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This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.

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An introduction to the embedding problem for probabilities on locally compact groups

Cited by 9 publications

References 27 publications

Addendum to the paper Affinely infinitely divisible distributions and the embedding problem

Addendum to the paper Affinely infinitely divisible distributions and the embedding problem

Affinely infinitely divisible distributions and the embedding problem

Explicit representation of roots on p-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

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