Transforming spatial point processes into Poisson processes
type: Item
instance of: scholarly article
title: Transforming spatial point processes into Poisson processes
zbMATH Open document ID: 0962.60029
DOI: 10.1016/S0304-4149(98)00098-2
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: Let \(N\) be a simple point process in the quadrant \({\mathbb R}_+^2\). The author shows that if the `coordinate sectional' intensities (\(\lambda _1(\omega ,s,t),\lambda _2(\omega ,s,t)\)) exist and if with probability one, \(\int \lambda _1(\omega ,s,t)dt=\infty \) for all \(s>0\), then there exists a family of \({\mathcal F}^1\)-predictable sets \(\{ D_z: z\in {\mathbb R}_+^2\}\) such that \(M:z\mapsto N(D_z)\) is a unit rate Poisson process. A possible generalization to higher dimensions is considered as well. These results generalize the original problem solved by Papangelou and Bremaud, whether the real line can be rescaled in such a way that a given point process on \(\mathbb R\) is transformed into a Poisson one.
reviewed by: Jan Rataj
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 1554310
zbMATH Keywords: point process
zbMATH Keywords: Poisson process
zbMATH Keywords: predictable set
zbMATH Keywords: stopping time
zbMATH Keywords: compensator
author: Frederic Paik Schoenberg
describes a project that uses: spatial
MaRDI profile type: Publication
cites work: Random time changes for multivariate counting processes
cites work: Poisson convergence in two dimensions with application to row and column exchangeable arrays
cites work: A simple proof of the multivariate random time change theorem for point processes
cites work: Stochastic integrals in the plane
cites work: Statistics for Spatial Data
cites work: Q5185878
cites work: Q5649230
cites work: Goodness-of-fit tests for the second moment funciton of a stationary multidimensional poisson process
cites work: Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales
cites work: A characterization of the spatial Poisson process and changing time
cites work: Random space change for multiparameter point processes
cites work: Integrability of Expected Increments of Point Processes and a Related Random Change of Scale
cites work: Q3862907
cites work: Q5515873
full work available at URL: s0304-4149(98)00098-2
OpenAlex ID: W2017501222
description: scientific article; zbMATH DE number 1554310