## Annals of Probability

### Particle Representations for Measure-Valued Population Models

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 27, Number 1 (1999), 166-205.

Dates
First available in Project Euclid: 29 May 2002

https://projecteuclid.org/euclid.aop/1022677258

Digital Object Identifier
doi:10.1214/aop/1022677258

Mathematical Reviews number (MathSciNet)
MR1681126

Zentralblatt MATH identifier
0956.60081

#### Citation

Donnelly, Peter; Kurtz, Thomas G. Particle Representations for Measure-Valued Population Models. Ann. Probab. 27 (1999), no. 1, 166--205. doi:10.1214/aop/1022677258. https://projecteuclid.org/euclid.aop/1022677258