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January 1999 Particle Representations for Measure-Valued Population Models
Peter Donnelly, Thomas G. Kurtz
Ann. Probab. 27(1): 166-205 (January 1999). DOI: 10.1214/aop/1022677258


Models of populations in which a type or location, represented by a point in a metric space $E$, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measure-valued processes are obtained as infinite population limits for a large class of neutral population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population and the de Finetti measures associated with an $E^{\infty}$ -valued particle model $X=(X_1, X_2\ldots)$ such that, for each $t \geq 0,(X_1(t),X_2(t),\ldots)$ is exchangeable. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both neutral Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of the Dawson-Perkins historical process and Perkins’s historical stochastic integral can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behavior are described.


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Peter Donnelly. Thomas G. Kurtz. "Particle Representations for Measure-Valued Population Models." Ann. Probab. 27 (1) 166 - 205, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0956.60081
MathSciNet: MR1681126
Digital Object Identifier: 10.1214/aop/1022677258

Primary: 60J25 , 60J70 , 60J80 , 60K35 , 92D10

Keywords: Coalescent , Conditioning , Dawson-Watanabe process , exchangeability , Fleming-Viot process , genealogical processes , Historical process , measure-valued diffusion , Superprocess

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 1 • January 1999
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