ON POISSON APPROXIMATIONS FOR THE EWENS SAMPLING FORMULA WHEN THE MUTATION PARAMETER GROWS WITH THE SAMPLE SIZE

• 成果类型: Article
• 署名作者: Tsukuda, Koji
• 署名单位: University of Tokyo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1433
• 发表日期: 2019
• 页码: 1188-1232
• 关键词: divide-and-conquer; logarithmic combinatorial; LIMIT-THEOREMS

摘要

The Ewens sampling formula was first introduced in the context of population genetics by Warren John Ewens in 1972, and has appeared in a lot of other scientific fields. There are abundant approximation results associated with the Ewens sampling formula especially when one of the parameters, the sample size n or the mutation parameter theta which denotes the scaled mutation rate, tends to infinity while the other is fixed. By contrast, the case that theta grows with n has been considered in a relatively small number of works, although this asymptotic setup is also natural. In this paper, when theta grows with n, we advance the study concerning the asymptotic properties of the total number of alleles and of the component counts in the allelic partition assuming the Ewens sampling formula, from the viewpoint of Poisson approximations. Specifically, the main contributions of this paper are deriving Poisson approximations of the total number of alleles, an independent process approximation of small component counts, and functional central limit theorems, under the asymptotic regime that both n and theta tend to infinity.