Probabilities on the Heisenberg group: limit theorems and Brownian motion
type: Item
instance of: scholarly article
title: Probabilities on the Heisenberg group: limit theorems and Brownian motion
zbMATH Open document ID: 0870.60007
author: Daniel Neuenschwander
published in: Lecture Notes in Mathematics
publication date: 1996-09-15T00:00:00Z
review text: After certain aspects of probability theory had been unified within the setting of general locally compact groups the main engagement of researchers working in the field changed to deepening the available results for special classes of groups. It was not surprising that this new development started in France where Bourbaki's spirit of utmost generality had somewhat faded in favor of a more problem-oriented trend of investigation. More precisely, in the succession of the monographs of \textit{U. Grenander} [``Probabilities on algebraic structures'' (1963; Zbl 0131.34804)], \textit{K. R. Parthasarathy} [``Probability measures on metric spaces'' (1967; Zbl 0153.19101)] and the reviewer [``Probability measures on locally compact groups'' (1977; Zbl 0376.60002)] the structurally oriented probabilists were able to help themselves to rich collections of particular examples as they were contained in the Lecture Notes of \textit{Y. Guivarc'h}, \textit{M. Keane} and \textit{B. Roynette} [``Marches aléatoires sur les groupes de Lie'' (1977; Zbl 0367.60081)] and \textit{P. Diaconis} [``Group representations in probability and statistics'' (1988; Zbl 0695.60012)] on random walks on Lie groups and on group representations in probability and statistics, respectively. Now, more than 20 years after the appearance of the reviewer's monograph up-to-date treatments of parts of the subject are highly desirable. The book under review deals with probabilities on the Heisenberg group \(\mathbb{H}\), in the not too far future we shall have access to a book by \textit{W. Hazod} and the late \textit{E. Siebert} on stable convolution semigroups on groups and vector spaces. The present Lecture Notes have been completed in the course of the author's ``Habilitation'' at the University of Bern. They are conceived expositorily but contain a large portion of original work accomplished by the author since his ``Dissertation'' in 1991. The author masters a great deal of literature on only 139 pages an…
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 60Fxx
zbMATH DE Number: 926908
zbMATH Keywords: probability measure on groups
zbMATH Keywords: Brownian motion on the Heisenberg group
zbMATH Keywords: laws of the iterated logarithm
zbMATH Keywords: two-series theorem
reviewed by: Herbert Heyer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0094029
OpenAlex ID: W2500078086
DOI: 10.1007/BFB0094029
description: scientific article; zbMATH DE number 926908