Two-parameter family of infinite-dimensional diffusions on the Kingman simplex

Abstract

We construct a two-parameter family of diffusion processes X α,θ on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers.For α = 0, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981) in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present paper, we extend Ethier and Kurtz's main results to the two-parameter case. Namely, we show that the (two-parameter) PoissonDirichlet distribution PD(α, θ) is the unique stationary distribution for the process X α,θ and that the process is ergodic and reversible with respect to PD(α, θ). We also compute the spectrum of the generator of X α,θ . The Wright-Fisher diffusions on finite-dimensional simplices turn out to be special cases of X α,θ for certain degenerate parameter values.

Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion

“…It was introduced in [16] and further studied in [10]. The unique reversible measure is the two-parameter Poisson-Dirichlet distribution, simply denoted by μ α,θ , and is defined as follows.…”

The sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution

“…We also consider a generalized two-parameter infinitely-many-alleles diffusion model. The twoparameter diffusion X α,θ (t) is constructed by different means in [6] and [16]. Here we compute the probability generating function of random sampling from X α,θ (t), which can be seen as a two-parameter generalization of Theorem 4.3 of [9].…”

“…In [6] and [16] the authors used different methods to construct a two-parameter version of the IMNA model. Specifically, the two-parameter infinite-alleles diffusion is an infinitedimensional symmetric diffusion process taking values in ∇ ∞ with generator…”

Countable representation for infinite dimensional diffusions derived from the two-parameter Poisson-Dirichlet process

“…The remainder of the section is dedicated to prove the existence of a suitably defined limiting process, which will coincide with that in [18], and the weak convergence of the process of ranked frequencies. In the following section we will then show that the limiting process is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution.…”

“…The statement of Theorem 3.6 can be strengthened. From [18] it follows that the sample paths of Y (·) belong to C ∇ ∞ ([0, ∞)) almost surely. Then [3] (cf.…”