Introducing Results Associated with Poisson Approximations for the Ewens Sampling Formula with Large Parameters

Abstract

From the viewpoint of Poisson approximations, we introduce properties of the Ewens sam pling formula based on the paper Tsukuda (2019, The Annals of Applied Probability 29, no. 2, 1188–1232). The Ewens sampling formula is a basic distribution of random partitions, and has two parameters n ∈ ℕ and θ > 0. In this paper, considering a n-dimensional random variable Cⁿ = (C₁ⁿ, ..., C_nⁿ) following the Ewens sampling formula, we focus on asymptotic properties of its b-dimensional marginal C_bⁿ = (C₁ⁿ, ..., C_bⁿ) (b = 1, 2, ...) and the sum K_n = ∑_i=1ⁿC_iⁿ when two parameters simultaneously increase.