The Annals of Applied Probability

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

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

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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.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 1 (2002), 101-124.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015961157

Digital Object Identifier
doi:10.1214/aoap/1015961157

Mathematical Reviews number (MathSciNet)
MR1890058

Zentralblatt MATH identifier
1010.62101

Subjects
Primary: 62J70 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60F05: Central limit and other weak theorems

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

Citation

Joyce, Paul; Krone, Stephen M.; Kurtz, Thomas G. Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula. Ann. Appl. Probab. 12 (2002), no. 1, 101--124. doi:10.1214/aoap/1015961157. https://projecteuclid.org/euclid.aoap/1015961157


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  • MOSCOW, IDAHO 83844-1103 E-MAIL: joyce@uidaho.edu krone@uidaho.edu T. G. KURTZ DEPARTMENTS OF MATHEMATICS AND STATISTICS UNIVERSITY OF WISCONSIN
  • MADISON, WISCONSIN 53706-1388 E-MAIL: kurtz@math.wisc.edu