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Wright-Fisher construction of the two-parameter Poisson-Dirichlet diffusion

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Constantini, Cristina, De Blasi, Pierpaolo, Ethier, Stewart N., Ruggiero, Matteo and Spanò, Dario (2017) Wright-Fisher construction of the two-parameter Poisson-Dirichlet diffusion. Annals of Applied Probability, 27 (3). pp. 1923-1950. doi:10.1214/16-AAP1252

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Official URL: http://doi.org/10.1214/16-AAP1252

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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 → ∞. Moreover, the descending order statistics of the K-dimensional diffusion converge in distribution to the two-parameter Poisson–Dirichlet diffusion as K → ∞. 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”, i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Diffusion processes, Distribution (Probability theory)
Journal or Publication Title: Annals of Applied Probability
Publisher: Institute of Mathematical Statistics
ISSN: 1050-5164
Official Date: 19 July 2017
Dates:
DateEvent
19 July 2017Published
18 October 2016Accepted
Volume: 27
Number: 3
Page Range: pp. 1923-1950
DOI: 10.1214/16-AAP1252
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
StG “N-BNP” 306406H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663
209632Simons Foundationhttp://dx.doi.org/10.13039/100000893
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