Gaussian limits associated with the Poisson-Dirichlet distribution and the Ewens sampling formula

• 成果类型: Article
• 署名作者: Joyce, P; Krone, SM; Kurtz, TG
• 署名单位: University of Idaho; University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 101-124
• 关键词: convergence; size

摘要

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.