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June 2017 Wright–Fisher construction of the two-parameter Poisson–Dirichlet diffusion
Cristina Costantini, Pierpaolo De Blasi, Stewart N. Ethier, Matteo Ruggiero, Dario Spanò
Ann. Appl. Probab. 27(3): 1923-1950 (June 2017). DOI: 10.1214/16-AAP1252

Abstract

The two-parameter Poisson–Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman’s one-parameter Poisson–Dirichlet distribution and to certain Fleming–Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here, we shed some light on these dynamics by formulating a $K$-allele Wright–Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $N\to\infty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson–Dirichlet diffusion as $K\to\infty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove a priori that in the limit there is no “loss of mass”, that is, that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.

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Cristina Costantini. Pierpaolo De Blasi. Stewart N. Ethier. Matteo Ruggiero. Dario Spanò. "Wright–Fisher construction of the two-parameter Poisson–Dirichlet diffusion." Ann. Appl. Probab. 27 (3) 1923 - 1950, June 2017. https://doi.org/10.1214/16-AAP1252

Information

Received: 1 January 2016; Revised: 1 August 2016; Published: June 2017
First available in Project Euclid: 19 July 2017

zbMATH: 1370.92091
MathSciNet: MR3678488
Digital Object Identifier: 10.1214/16-AAP1252

Subjects:
Primary: 92D25
Secondary: 60F17, 60G57, 60J60

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 3 • June 2017
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