Randomizable point systems
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.
Kallenberg's example of a stationary non-Coxian line process
scientific article; zbMATH DE number 4034741