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Paper No. 14-11

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P De Blash, M Ruggiero and D Spano

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

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 infinite-population construction, with infinitely-many species, of the twoparameter infinite-dimensional diffusion. The construction 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.

Infinite-alleles model, Poisson{Dirichlet distribution, reinforcement, immigration, Wright{Fisher model.