Inhomogeneous Wright-Fisher construction of two-parameter Poisson-Dirichlet diffusions

TL;DR

This paper provides a finite-population Wright-Fisher construction for the two-parameter Poisson-Dirichlet diffusion, elucidating how the additional parameter influences population clustering through inhomogeneous immigration dynamics.

Contribution

It introduces a novel finite-population Wright-Fisher model with inhomogeneous immigration to understand the two-parameter Poisson-Dirichlet diffusion.

Findings

Finite-population construction of the diffusion process.

Detailed analysis of inhomogeneous immigration dynamics.

Balance between reinforcement and redistribution effects.

Abstract

The recently introduced two-parameter Poisson-Dirichlet diffusion extends the infinitely-many-neutral-alleles model, related to Kingman's distribution and to Fleming-Viot processes. The role of 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 providing a finite-population construction, with finitely-many species, of the two-parameter infinite-dimensional diffusion. The costruction is obtained in terms of Wright-Fisher chains that feature a classical symmetric mutation mechanism and a frequency-dependent immigration, whose inhomogeneity is investigated in detail. The local immigration dynamics are built upon an underlying array of Bernoulli trials and can be described by means of a dartboard experiment and a rank-dependent type distribution. These involve a delicate balance between reinforcement and redistributive effects, among the current species abundances, for the convergence to hold.