Distribution of the particles of a critical branching Wiener process
type: Item
instance of: scholarly article
title: Distribution of the particles of a critical branching Wiener process
zbMATH Open document ID: 0859.60077
DOI: 10.2307/3318569
published in: Bernoulli
publication date: 1996-06-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1193758790
review text: In this model, a single particle starts at the origin and executes a Wiener process \(W(t)\) in \(\mathbb{R}^d\) until time \(t=1\). It is then replaced by \(Y\geq 0\) offspring which independently perform Wiener processes, starting from the point \(W(1)\), and then after another time unit are each replaced by more offspring with the distribution of \(Y\), and so on. It is assumed that \(Y\) has mean \(1\). The main result concerns the standardized distribution function \(F_T({\mathbf x})\), \({\mathbf x}\in\mathbb{R}^d\), conditional on non-extinction, of the locations of particles living at time \(T=1,2,\dots\), and states that any absolutely continuous distribution function \(F({\mathbf x})\) with bounded density will be well approximated by \(F_T({\mathbf x})\) with positive probability, for all sufficiently large \(T\). The model has previously been treated by the author [``Random walks of infinitely many particles'' (1994; Zbl 0841.60053)].
reviewed by: M. P. Quine
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 887273
zbMATH Keywords: critical branching Wiener process
zbMATH Keywords: empirical distribution of particle locations
zbMATH Keywords: conditional limit theorems
author: Pál Révész
MaRDI profile type: Publication
OpenAlex ID: W4250677071
description: scientific article; zbMATH DE number 887273