Harmonic Analysis of Probability Measures on Hypergroups

Bloom, Walter R.; Heyer, Herbert

doi:10.1515/9783110877595

Cited by 383 publications

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“…As shown in [BH,VI,V2,Zl,Z2], they provide a useful tool in probability theory. In many cases, they can be constructed as directional derivatives in the dual space at the identity character 1.…”

Section: Moment Functionsmentioning

confidence: 99%

Limit theorems for isotropic random walks on triangle buildings

Lindlbauer

Voit²

2002

J. Aust. Math. Soc.

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The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynomial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynomials.2000 Mathematics subject classification: primary 60B15; secondary 60F05, 60F15, 33D52, 20E42, 43A62.

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Section: Moment Functionsmentioning

confidence: 99%

Limit theorems for isotropic random walks on triangle buildings

Lindlbauer

Voit²

2002

J. Aust. Math. Soc.

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“…Since ν (1) ∈ P log + , for a iid sequence (Y j ) of random variables with distribution ν (1) we obtain |τ j (Y j ) | < ∞ a.s. (Proposition 3.7 (3.1)). τ satisfies the growth condition (2.1),…”

Section: Multiple τ -Decomposability Of Probabilitiesmentioning

confidence: 99%

“…If we stay in the locally compact set-up, hypergroups are natural candidates. For the background the reader is referred to the monograph of W. Bloom and H. Heyer [1]. Recently M. Voit [34] and M. Rösler [24] investigated new classes of hypergroup structures on matrix cones with 'group like' properties.…”

Section: Appendix: Multiple Decomposability On Matrix-cone Hypergroupsmentioning

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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“…In this way one obtains continuous families of commutative hypergroup structures (X; * p ) with these special functions are characters, which generalize discrete families of double coset hypergroups; see [2] and [13] for the theory and notion of hypergroups. The hypergroup convolution * p on X allows to introduce the notion of random walks on (X, * ) by saying that a (time-homogeneous) Markov (Z p n ) n≥0 is a random walk on (X; * p ) with law ν ∈ M 1 (X) if…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Voit

2008

Infinite Dimensional Harmonic Analysis IV

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Harmonic Analysis of Probability Measures on Hypergroups

Cited by 383 publications

References 0 publications

Limit theorems for isotropic random walks on triangle buildings

Limit theorems for isotropic random walks on triangle buildings

Multiple decomposability of probabilities on contractible locally compact groups

Limit theorems for radial random walks on homogeneous spaces with growing dimensions

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