On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size

Abstract

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.