An occupation time approach for convergence of measure-valued processes, and the death process of a branching system
type: Item
instance of: scholarly article
title: An occupation time approach for convergence of measure-valued processes, and the death process of a branching system
zbMATH Open document ID: 0808.60075
DOI: 10.1016/0167-7152(94)90058-2
published in: Statistics \& Probability Letters
publication date: 1994-11-06T00:00:00Z
review text: Let \({\mathcal M}_ p (R^ d)\) be a space of Radon measures \(\mu\) on \(R^ d\) such that \(\int (1 + | x|^ 2)^{-p} d\mu < \infty\) and let \(D(R_ +, {\mathcal M}_ p (R^ d))\) be a space of right-continuous with left limits functions from \(R_ +\) into \({\mathcal M}_ p (R^ d)\), endowed with the Skorokhod topology. The \((d,\alpha, \beta, b)\)- superprocess \(X = \{X(t), t \geq 0\}\) is a time-homogeneous Markov process with paths in \(D(R_ +, {\mathcal M}_ p (R^ d))\) whose transition Laplace functional is given by some nonlinear differential equation. It is constructed a family of time and \(n\) dependent Markov branching particle systems on \(R^ d\), and then particle processes \(N^ n = \{N^ n (t)\), \(t \geq 0\}\) of the particle systems which are the sum of the delta-functions of locations of the particles present at time \(t\) and death processes \(D^ n = \{D^ n (t)\), \(t \geq 0\}\) of the particle systems which are the sum of the delta-functions of locations of the particles that have died up to time \(t\). Then it is defined \(X^ n = n^{-1} N^ n\) and \(G^ n = m_ n ^{-1} D^ n\), where \(m_ n = cn^{1 + \beta}\) (it is given particle mass \(n^{-1}\) and \(m_ n^ {- 1}\)). The main result is: If \(X^ n(0) \to X(0)\), then for each \(t > 0\) \[ \left(X^ n (t), \int_ 0^ t X^ n(s) ds,\;G^ n(t)\right)\to\left(X(t), \int_ 0^ t X(s) ds, G(t)\right) \text{ as } n \to \infty, \] and it is given the Laplace functional of the limit as a solution of some nonlinear differential equation.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 681919
zbMATH Keywords: measure-valued process
zbMATH Keywords: weak convergence
zbMATH Keywords: branching particle system
zbMATH Keywords: superprocess
zbMATH Keywords: occupation time
zbMATH Keywords: death process
author: Luis G. Gorostiza
author: José Alfredo López-Mimbela
reviewed by: Valentin Topchii
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)90058-2
OpenAlex ID: W2040190485
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description: scientific article; zbMATH DE number 681919