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

TL;DR

This paper constructs a Wright-Fisher model with a specific migration mechanism that converges to the two-parameter Poisson-Dirichlet diffusion, providing insights into how the second parameter influences population clustering.

Contribution

It introduces a finite-population Wright-Fisher model that converges to the two-parameter Poisson-Dirichlet diffusion, elucidating the role of the second parameter in population dynamics.

Findings

Convergence of scaled finite models to the infinite-dimensional diffusion.

Descending order statistics converge to the Poisson-Dirichlet distribution.

The migration mechanism balances reinforcement and redistribution effects.

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 \textit{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.