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{1015961157,
author = {Joyce, Paul and Krone, Stephen M. and Kurtz, Thomas G.},
title = {Gaussian Limits Associated with the Poisson-Dirichlet Distribution
and the Ewens Sampling Formula},
journal = {Ann. Appl. Probab.},
volume = {12},
number = {1},
year = {2002},
pages = { 101-124},
language = {en},
url = {http://dml.mathdoc.fr/item/1015961157}
}
Joyce, Paul; Krone, Stephen M.; Kurtz, Thomas G. Gaussian Limits Associated with the Poisson-Dirichlet Distribution
and the Ewens Sampling Formula. Ann. Appl. Probab., Tome 12 (2002) no. 1, pp. 101-124. http://gdmltest.u-ga.fr/item/1015961157/