Central limit theorem and infinitely divisible probabilities associated with partial differential operators
type: Item
instance of: scholarly article
title: Central limit theorem and infinitely divisible probabilities associated with partial differential operators
zbMATH Open document ID: 0834.60012
DOI: 10.1007/BF02218040
author: Mohamed Sifi
published in: Journal of Theoretical Probability
publication date: 1995-11-22T00:00:00Z
review text: The spherical functions of the Gelfand pair \((U(n,1), U(n))\) are given in terms of two-dimensional Jacobi functions \(\varphi^\alpha_{\lambda, \mu}\) with \(\alpha = \alpha (n)\); this was shown in a series of classical papers of Flensted-Jensen and Koornwinder. Motivated by the associated double coset hypergroups \(U(n,1)//U(n)\), the author studies two- dimensional commutative hypergroup structures whose characters are two- dimensional Jacobi functions \(\varphi^\alpha_{\lambda, \mu}\) with \(\alpha \in \mathbb{R}\), \(\alpha \geq 0\). The main purpose is to present some results in algebraic probability theory for these and related hypergroups. Unfortunately, the associated convolutions are not defined properly for some cases which leads to some incorrect statements about these convolutions. To describe this, consider the underlying spaces and the associated convolutions. By Theorem 3.1 of \textit{M. Flensted-Jensen} [Math. Ann. 228, 65-92 (1977; Zbl 0331.43013)], there is a covering group \(\widetilde G\) of \(U(n,1)\) containing \(U(n)\) in a natural way such that \(\widetilde G//U(n)\) can be regarded as a commutative hypergroup on \([0, \infty [\times \mathbb{R}\) with a convolution given via product formulas for two- dimensional Jacobi functions. It was observed by \textit{K. Trimeche} [J. Math. Pures Appl., IX. Sér. 70, 1-73 (1991; Zbl 0671.35004)] that these formulas hold for all \(\alpha \in \mathbb{R}\), \(\alpha \geq 0\). They also lead to commutative hypergroups on \(K : = [0, \infty [\times \mathbb{R}\) such that \(\{0\} \times \mathbb{R}\) is a subgroup isomorphic to \((\mathbb{R},+)\). In this way, for each \(\lambda > 0\), the set \(G_\lambda : = \{0\} \times \lambda \mathbb{Z}\) is a subgroup of \(K\), and the quotient \(K/G_\lambda\) carries a unique commutative quotient hypergroup structure isomorphic with an appropriate hypergroup on \([0, \infty [\times {\mathbf T}\) or \(\{z \in \mathbb{C} : |z |\geq 1\}\). The author claims to construct con…
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 806425
zbMATH Keywords: central limit theorems
zbMATH Keywords: infinitely divisible measures
zbMATH Keywords: Gelfand pair
zbMATH Keywords: commutative hypergroup structures
zbMATH Keywords: commutative quotient hypergroup structure
zbMATH Keywords: convolution semigroups
zbMATH Keywords: Lévy-Khinchin representation
zbMATH Keywords: Lévy-Khinchin formula
reviewed by: Michael Voit
MaRDI profile type: Publication
cites work: Convolution semigroups and resolvent families of measures on hypergroups
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cites work: The Spectral Theory of Bounded Functions
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description: scientific article; zbMATH DE number 806425