Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula

Abstract

In this paper we consider large $\theta$ approximations for the stationary distribution of the neutral infinite alleles model as described by the the Poisson–Dirichlet distribution with parameter $\theta$. We prove a variety of Gaussian limit theorems for functions of the population frequencies as the mutation rate $\theta$ goes to infinity. In particular, we show that if a sample of size $n$ is drawn from a population described by the Poisson–Dirichlet distribution, then the conditional probability of a particular sample configuration is asymptotically normal with mean and variance determined by the Ewens sampling formula. The asymptotic normality of the conditional sampling distribution is somewhat surprising since it is a fairly complicated function of the population frequencies. Along the way, we also prove an invariance principle giving weak convergence at the process level for powers of the size-biased allele frequencies.