Gibbs point process approximation: total variation bounds using Stein's method
type: Item
instance of: scholarly article
title: Gibbs point process approximation: total variation bounds using Stein's method
zbMATH Open document ID: 1322.60060
DOI: 10.1214/13-AOP895
author: Dominic Schuhmacher
author: Kaspar Stucki
published in: The Annals of Probability
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.3096
full work available at URL: https://projecteuclid.org/euclid.aop/1409319470
review text: Point processes (fields) on a compact metric space \(\mathcal X\) are considered. Such a point process is determined by a probability measure on the set \(\mathcal N\) of all finite subsets of this space. This probability measure depends on their family of conditional probabilities. A class of Gibbs processes is defined in terms of a partial ordering on the set \(\mathcal N\). The best known Gibbs process is constructed on the base of pairwise interaction of points from its realizations. This interaction is determined by some non-negative function \(\varphi\) on \(\mathcal X \times \mathcal X\). The Gibbs process is said to be a hard core process if there exists some \(a > 0\) such that \(\varphi (x_1 , x_2 ) = 1\) when \(\text{dist}(x_1 , x_2 ) \leq a\), and \(\varphi(x_1 , x_2 ) = 0\) when \(\text{dist}(x_1 , x_2 ) > a\). Another example of a Gibbs process is called Lennard-Jones process. It has an exponentially decreasing function \(\varphi\) as a function of distance between its arguments. A distance on \(\mathcal N\) and the corresponding distance on the set of measures on \(\mathcal N\) are defined. The distance between two Gibbs distributions is proved to be estimated from above in terms of a distance between their conditional intensities. These estimates permit to determine an approximation of general type Gibbs processes with the help of more investigated classes (for example, hard core Gibbs processes, or Lennard-Jones processes). The obtained results are applied for the solution of the coupling problem for two processes of birth and death. The main method of investigation used in the paper is a method of Stein with the proofs are being reduced to the solution of some integral-differential equation.
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 6363037
zbMATH Keywords: Gibbs point processes
zbMATH Keywords: total variation distance
zbMATH Keywords: Stein's method
zbMATH Keywords: conditional intensity
zbMATH Keywords: pairwise interaction process
zbMATH Keywords: birth-death process
reviewed by: B. P. Harlamov
MaRDI profile type: Publication
arXiv ID: 1207.3096
cites work: Area-interaction point processes
cites work: Q3810611
cites work: Stein's method and point process approximation
cites work: An Introduction to Stein's Method
cites work: Compound Poisson process approximation.
cites work: Q4092750
cites work: Point process limits of lattice processes
cites work: Stein's method, Palm theory and Poisson process approximation.
cites work: Entropy decay for interacting systems via the Bochner-Bakry-Émery approach
cites work: An Introduction to the Theory of Point Processes
cites work: Q3721531
cites work: Probability and random processes.
cites work: Q4114580
cites work: Q2774021
cites work: Glauber dynamics of continuous particle systems
cites work: GIBBS MEASURES OF CONTINUOUS SYSTEMS: AN ANALYTIC APPROACH
cites work: On the rate of convergence of spatial birth-and-death processes
cites work: Q4807532
cites work: Q4158279
cites work: Q5564295
cites work: Distance estimates for dependent thinnings of point processes with densities
cites work: Q4404195
cites work: Bounds for the Probability Generating Functional of a Gibbs Point Process
cites work: On the asymptotics of locally dependent point processes
OpenAlex ID: W1964263897
description: scientific article; zbMATH DE number 6363037