Randomizable point systems
type: Item
instance of: scholarly article
title: Randomizable point systems
zbMATH Open document ID: 0635.60013
DOI: 10.1007/BF00580823
author: G. S. Sukiasian
published in: Acta Applicandae Mathematicae
publication date: 1987-01-01T00:00:00Z
review text: Let G be a group of transformations of the space \({\mathbb{R}}^ n.\) The author calls a point system \(\omega\) (i.e. a subset of \({\mathbb{R}}^ n \)with no accumulation points) G-randomizable, if there exists a G- invariant probability measure on the set of images \(\{\) \(g\omega\) :g\(\in G\}\). A nontrivial instance of this occurs in Siegel's theorem, which is at the heart of Kallenberg's example of a stationary non-Coxian line process in the plane [\textit{O. Kallenberg}, Math. Proc. Camb. Philos. Soc. 82, 301-307 (1977; Zbl 0401.60050)]: it asserts that the integer lattice of \({\mathbb{R}}^ n \)is \({\mathcal A}_ n\)-randomizable, if \({\mathcal A}_ n\) denotes the group of affine transformations of \({\mathbb{R}}^ n \)which preserve the Lebesgue measure. The present paper gives a survey of the author's recent work on randomizable point systems [e.g.: Izv. Akad. Nauk Arm. SSR, Mat. 20, No.4, 229-306 (1985; Zbl 0598.60047)]. In the last part of the paper the author applies parabolic inversions to Kallenberg's non-Coxian line process, thereby generating further examples of such processes.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 4034741
zbMATH Keywords: Siegel's theorem
zbMATH Keywords: Kallenberg's example of a stationary non-Coxian line process
zbMATH Keywords: randomizable point systems
zbMATH Keywords: parabolic inversions
reviewed by: F. Papangelou
MaRDI profile type: Publication
cites work: Q3223604
cites work: Q4066300
cites work: A counterexample to R. Davidson's conjecture on line processes
cites work: An explicit description of Kallenberg's lattice type point process
cites work: Q5669004
cites work: Q5601290
full work available at URL: https://doi.org/10.1007/bf00580823
OpenAlex ID: W4233614497
description: scientific article; zbMATH DE number 4034741