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

Cristina Costantini, Pierpaolo De Blasi, Stewart N. Ethier, Matteo Ruggiero, Dario Spano

The two-parameter Poisson-Dirichlet diffusion, recently introduced 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 parent-independent mutation pattern and a specific state-dependent immigration kernel. 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 immigration kernel 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 finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.