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February 2002 Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula
Paul Joyce, Stephen M. Krone, Thomas G. Kurtz
Ann. Appl. Probab. 12(1): 101-124 (February 2002). DOI: 10.1214/aoap/1015961157


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.


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Paul Joyce. Stephen M. Krone. Thomas G. Kurtz. "Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula." Ann. Appl. Probab. 12 (1) 101 - 124, February 2002.


Published: February 2002
First available in Project Euclid: 12 March 2002

zbMATH: 1010.62101
MathSciNet: MR1890058
Digital Object Identifier: 10.1214/aoap/1015961157

Primary: 62J70 , 92D10
Secondary: 60F05

Keywords: Ewens sampling formula , Gaussian process , GEM , mutation rate , neutral infinite alleles model , Poisson-Dirichlet , sampling distribution

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 1 • February 2002
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