Probability Measures on Locally Compact Groups

Heyer, Herbert

doi:10.1007/978-3-642-66706-0

Cited by 513 publications

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“…For any second countable locally compact group G, the impli- We have for N, M > 0: µ N +M = µ N λ N,N +M . According to the shift compactness theorem [22], III, theorem 2.1, 2.2, [10], theorem 1.2.21, the set {λ N,N +M } is relatively compact. Let L denote the set of accumulation points (for N → ∞).…”

Section: Convergence Of Convolution Products On Contractible Groupsmentioning

confidence: 99%

“…I.e., λ = ε e , and L = {ε e }. According to [10], 2.2.16, (i) follows. Assume (iii), and assume further that G is aperiodic.…”

Section: Convergence Of Convolution Products On Contractible Groupsmentioning

confidence: 99%

“…Recall that for non-Abelian groups, generating functionals of continuous convolution semigroups play the role of logarithms of Fourier transforms, the second characteristic functions. For details the reader is referred to, e.g., [10], Ch IV, § 2.0, [7], [29], [3].…”

Section: Introductionmentioning

confidence: 99%

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Multiple decomposability of probabilities on contractible locally compact groups

Hazod

Kosfeld

2013

Proc Math Sci

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Abstract. Operator decomposable probabilities on vector spaces -generalizing (semi-)stable and self-decomposable laws -are well known. More specific concepts, multiple operator decomposable laws, generalizing the nested Urbanik classes in the case of selfdecomposability, were investigated in fundamental papers by Maejima et al. [16], [17] resp. by Maejima and R. Shah [18] for real resp. p-adic vector spaces. For locally compact groups, decomposability properties were studied by R. Shah [27], Raja [23], see also [6]. Here we are concerned with multiple decomposability on locally compact groups and generalize -as far as possible -the results in [16], [18]. In fact, as it turned out that contraction properties play an essential role, hence throughout we concentrate on contractible locally compact groups.

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Section: Convergence Of Convolution Products On Contractible Groupsmentioning

confidence: 99%

“…I.e., λ = ε e , and L = {ε e }. According to [10], 2.2.16, (i) follows. Assume (iii), and assume further that G is aperiodic.…”

Section: Convergence Of Convolution Products On Contractible Groupsmentioning

confidence: 99%

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Multiple decomposability of probabilities on contractible locally compact groups

Hazod

Kosfeld

2013

Proc Math Sci

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“…The inspiration for writing up previous thoughts on the subject going back to the author's monographs [4] and [5] is due to D. Applebaum who in his upcoming book [1] points out that the harmonic analysis of compact groups still lacks a precise analogue of Bochner's theorem.…”

Section: Introductionmentioning

confidence: 99%

A Bochner-type representation of positive definite mappings on the dual of a compact group

Heyer¹

2013

COSA

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Abstract. For an arbitrary compact group G with dual space Σ(G) a Bochner-type bijection is established between positive definite mappings on Σ(G) and central bounded measures on G. This bijection defined by the generalized Fourier transformation is based on the comparative study of three kinds of function algebras: the coefficient algebra, the algebra of convergent Fourier series, and the central Fourier algebra of G.

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“…Sections 2 and 3 contain preliminaries on the method of Gelfand pairs and spherical functions along the standard references Dieudonne [3] and Bougerol [2]. For the theory of probability measures on locally compact groups the reader is referred to parts of the author's monograph [6], It is the author's pleasure to pay tribute to K. Hunzinger who in his Diploma thesis struggled successfully with some of the sources. § 2.…”

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confidence: 99%

Infinitely divisible probability measures on a discrete Gelfand pair

Heyer¹

1989

Nagoya Mathematical Journal

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Infinitely divisible probability measures on discrete spaces admitting a commutative convolution have been studied in various frameworks. For generalized convolutions related to Delphic structures an important contribution was made by Gilewski and Urbanik in [5]. In [11] Schwartz based his detailed analysis on convolutions arising from orthogonal series. Both of these approaches can be included into the framework of discrete hypergroups chosen f.e. by Gallardo and Gebuhrer in [4]. The main result common to these sources is the fact that, roughly speaking, all infinitely divisible probability measures are of Poisson type. Although the double coset spaces of Gelfand pairs are commutative hypergroups admitting an extended harmonic anafysis, the analytic methods developed in the theory of Gelfand pairs (see Dieudonné [3] and Heyer [7]) provide a more direct access to specific results like the characterization of divisible (idempotent, infinitely divisible) measures. For discrete Gelfand pairs (and their double coset spaces) Letac and his school have achieved remarkable results (see Letac [8], [9] and the references therein). The still unpublished thesis [1] of S. Ben Mansoor contains much information about divisible probability measures on cubes. In our exposition we reprove Ben Mansoor’s main theorem 3.5.4 with additional care, and discuss the special case of the ra-dimensional cube which was the basic object of study already in Letac, Takács [10]. It turns out that the main obstacle to be overcome in establishing the Poisson representation of infinitely divisible probability measures is the presence of idempotent factors, a problem that has been excluded f.e. in the work [4] of Gallardo and Gebuhrer.

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Probability Measures on Locally Compact Groups

Cited by 513 publications

References 0 publications

Multiple decomposability of probabilities on contractible locally compact groups

Multiple decomposability of probabilities on contractible locally compact groups

A Bochner-type representation of positive definite mappings on the dual of a compact group

Infinitely divisible probability measures on a discrete Gelfand pair

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