Extreme sizes in the Gibbs-type exchangeable random partitions

TL;DR

This paper investigates the asymptotic behavior of extreme sizes in Gibbs-type exchangeable random partitions, providing explicit formulas and analyzing specific models like Ewens-Pitman and Gnedin partitions.

Contribution

It introduces a combinatorial approach using partial Bell polynomials to analyze the asymptotics of extreme sizes in Gibbs partitions, including new results for specific models.

Findings

Explicit asymptotic formulas for extreme sizes in Gibbs partitions

Analysis of Ewens-Pitman and Gnedin partitions

Formulas for associated partial Bell polynomials

Abstract

Gibbs-type exchangeable random partitions, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially, Ewens-Pitman partition, which is the sample from the Poisson-Dirichlet process and has been discussed from rather model-specific viewpoints, and a random partition which was recently introduced by Gnedin, are discussed in the details. As by-products, some formulas for the associated partial Bell polynomials are presented.