## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 3 (2017), 1923-1950.

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

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

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1923-1950.

**Dates**

Received: January 2016

Revised: August 2016

First available in Project Euclid: 19 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1500451247

**Digital Object Identifier**

doi:10.1214/16-AAP1252

**Mathematical Reviews number (MathSciNet)**

MR3678488

**Zentralblatt MATH identifier**

1370.92091

**Subjects**

Primary: 92D25: Population dynamics (general)

Secondary: 60J60: Diffusion processes [See also 58J65] 60G57: Random measures 60F17: Functional limit theorems; invariance principles

**Keywords**

Infinite-dimensional diffusion process two-parameter Poisson–Dirichlet distribution reinforcement migration Wright–Fisher model weak convergence

#### Citation

Costantini, Cristina; De Blasi, Pierpaolo; Ethier, Stewart N.; Ruggiero, Matteo; Spanò, Dario. Wright–Fisher construction of the two-parameter Poisson–Dirichlet diffusion. Ann. Appl. Probab. 27 (2017), no. 3, 1923--1950. doi:10.1214/16-AAP1252. https://projecteuclid.org/euclid.aoap/1500451247