Particle representations for measure-valued population models
type: Item
instance of: scholarly article
title: Particle representations for measure-valued population models
zbMATH Open document ID: 0956.60081
author: Peter Donnelly
author: Thomas G. Kurtz
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: The authors consider two continuous time models for the evolution of a finite population in which a type or location, represented by a point in a metric space \(E\), is associated with each individual. The chances of an individual replicating or dying do not depend on its type. In one model, the population is ordered by decreasing age and in the other, this order does not play a role. Measure-valued processes [see \textit{D. A. Dawson}, in: Ecole d'Été de probabilités de Saint Flour XXI-1991. Lect. Notes Math. 1541, 1-260 (1993; 799.60080)] are obtained as infinite population limits for a large class of population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of \textit{D. Dawson}'s and \textit{E. A. Perkins}'s historical process [Mem. Am. Math. Soc. 454 (1991; Zbl 0754.60062)]. \textit{E. Perkins}' historical stochastic integral [Probab. Theory Relat. Fields 94, No. 2, 189-245 (1992; Zbl 0767.60044)] can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behaviour are described.
reviewed by: P. R. Parthasarathy
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G52
zbMATH DE Number: 1359644
zbMATH Keywords: measure-valued diffusion
zbMATH Keywords: historical process
zbMATH Keywords: conditioning
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677258
OpenAlex ID: W2114411217
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DOI: 10.1214/AOP/1022677258
description: scientific article; zbMATH DE number 1359644