A Compound Poisson Perspective of Ewens–Pitman Sampling Model

Abstract

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ>−α, or α0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

Keywords

• Berry–Esseen type theorem
• Ewens–Pitman sampling model
• exchangeable random partitions
• log-series compound poisson sampling model
• Mittag–Leffler distribution function
• negative binomial compound poisson sampling model