Structural, continuity, and asymptotic properties of a branching particle system
type: Item
instance of: scholarly article
title: Structural, continuity, and asymptotic properties of a branching particle system
zbMATH Open document ID: 1189.60157
published in: Lithuanian Mathematical Journal
publication date: 2009-11-06T00:00:00Z
review text: Let \(W(\delta,q)\) be a Pólya-Aeppli distributed random variable with parameters \(\delta\in\mathbb{R}+\) and \(q\in (0,1)\), i.e., \(\log\,{\mathbf E}\exp\{sW(\delta, q)\}= \delta(e^s- 1)/(1/(1- q)- e^s)\) for \(s<-\log(1-q)\). Note that the range of this r.v. is \(\mathbb{Z}+\) and that the random sum of \(W(\delta,q)\) i.i.d. random variables, which follow a common zero-modified geometric law, is again Pólya-Aeppli distributed. These properties are particularly useful in the context of classical branching processes. Consider a Markov branching process \({{\mathfrak L}^{(\eta)}}_{\delta, t}\) started with the number \(W(\delta, \delta/(\delta+\eta))\) (with mean \(\eta\)) of particles at time \(t= 0\). Let the particles move on \(\mathbb{R}^d\), each undergoing critical binary branching after an exponentially distributed lifetime with mean \(1/\eta\). The particles are identical, their motions, lifetimes and branching independent of each other. Let \({\widetilde T^{(\eta)}}_\delta(t)\) denote the of particle alive at time \(t= 0\). Then, for each \(t> 0\), \({\widetilde L^{(\eta)}}_\delta(t)\) is distributed as \(W(2\delta/(2+\delta t), 2/(2+2\eta/\delta t))\). Now let the random measure \({{\mathfrak L}^{(\eta)}}_{\delta,0}\) at time \(t_0= 0\) represent the current state of a process that originated sometime in the past. Assume that the evolution mechanism from past to present was the same as it is for the future and that the number of particles alive at an admissible instant \(s< 0\) is Pólya-Aeppli distributed with parameters \(\Delta_s\) and \(Q_s\), say. Then the precise lower bound for the admissible times is \(-2/\delta\), and the parameter values at \(s\in(-2/\delta,0)\) are \(\Delta_s= 2\delta/(2+s\delta)\) and \(Q_s= \Delta_s/(\eta+\Delta_s)\). The process \(\{{\widetilde L^{(\eta)}}_\delta(s); s> -\delta/2\}\) must have originated from a Poisson field with mean \(1/\eta\) at \(s= -2/\delta\), and it has a realization continuous in mean square. Explici…
reviewed by: Heinrich Hering
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 5626712
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Athreya-Ney-type representation
zbMATH Keywords: Babel class
zbMATH Keywords: Bessel function
zbMATH Keywords: birth-death Markov process
zbMATH Keywords: branching-diffusing particle system
zbMATH Keywords: cluster structure
zbMATH Keywords: continuity of stochastic process in mean square
zbMATH Keywords: Dawson-Watanabe superprocess
zbMATH Keywords: Esscher transform
zbMATH Keywords: Jørgensen-type convergence
zbMATH Keywords: Laplace method
zbMATH Keywords: leading error term
zbMATH Keywords: Lévy process
zbMATH Keywords: local limit theorem
zbMATH Keywords: Poisson-exponential law
zbMATH Keywords: Poisson mixture
zbMATH Keywords: Pólya-Aeppli law
zbMATH Keywords: power-variance family
zbMATH Keywords: summation invariance
zbMATH Keywords: unit variance function
zbMATH Keywords: Yaglom-type exponential limit law
zbMATH Keywords: zero-modified geometric law
author: Kenneth J. Hochberg
author: Vladimir Vinogradov
MaRDI profile type: Publication
full work available at URL: s10986-009-9049-5
OpenAlex ID: W2081007011
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Structural, continuity, and asymptotic properties of a branching particle system (DOI)
DOI: 10.1007/S10986-009-9049-5