Semistable convolution semigroups on measurable and topological groups
type: Item
instance of: scholarly article
title: Semistable convolution semigroups on measurable and topological groups
zbMATH Open document ID: 0544.60021
author: Eberhard Siebert
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1984-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77229
review text: Semistable probability measures on the real line have been first considered by P. Lévy. The extension to operator-semistable laws on Euclidean spaces is due to R. Jajte. Far reaching generalizations to topological and measurable groups have been studied by \textit{A. Tortrat} [see e.g., Probability measures on groups, Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 452-466 (1982; Zbl 0514.60014)]. Following this approach we adopt in this paper the following definition: A convolution semigroup (c.s.g.) \((\mu_ t)_{t>0}\) of probability measures on a measurable group G is said to be (strictly) semistable with respect to a measurable homomorphism \(\delta\) of G and with coefficient \(c\in(0,1)\cup(1,\infty)\) if \(\delta(\mu_ t)=\mu_{ct}\) for all \(t>0.\) Several examples from the literature that fit into this framework are presented in section 1. In section 2 a rather general result on purity of semistable c.s.g. is proved (theorem 1): These semigroups respect certain band decompositions of the Banach lattice of bounded measures on G. On the one hand side this leads to the well known zero-one laws for semistable measures and on the other hand side to the result that semistable c.s.g. on locally compact groups are of pure Lebesgue type. In section 3 we show that under rather weak conditions a semistable c.s.g. is holomorphic (theorem 5). This extends a well known property of stable c.s.g. also due to the author [Holomorphic convolution semigroups on topological groups. Probability measures on groups VII. Proc. Conf., Oberwolfach 1983. Lect. Notes Math. 1064, 421-449 (1984)]. In the semistable case the proof is more complicated; one first has to establish the quasianalyticity of the c.s.g. (theorem 4). In section 4 the results of section 3 are specialized to Euclidean spaces. In particular the support structure of a semistable c.s.g. is investigated (theorem 7). Remark: It has been incorrectly stated in our paper that a topological Hausdorff group G is also …
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 3866269
zbMATH Keywords: holomorphicity
zbMATH Keywords: Semistable probability measures
zbMATH Keywords: operator-semistable laws
zbMATH Keywords: convolution semigroup
zbMATH Keywords: quasianalyticity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3866269