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

*Paul Joyce, Stephen M. Krone, and Thomas G. Kurtz*
**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 dertermined 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.