Extreme sizes in Gibbs-type exchangeable random partitions

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.

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

“…Corollary 5.10 yields that, under the assumption of Proposition 5.9, P(S 1 n = 1) ∼ e −θ , so if θ → ∞ then P(S 1 n = 1) → 1 and if θ → c < ∞ then P(S 1 n = 1) → e −c < 1. Note that when the Pitman sampling formula is considered, the shortest cycle length converges to 1 in probability except the Ewens sampling formula (see Mano (2017)).…”

“…Theorem 5.13 directly implies the following corollaries which represent properties of the shortest cycle length S n = S 1 n and the longest cycle length L n in a Ewens partition. These extreme sizes are of interest in the combinatorial context, see for instance Mano (2017).…”

Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

“…with the convention B 0,k (w · ) = δ 0,k . The author has defined associated versions of the partial Bell polynomials [10]. They are generalizations of the partial Bell polynomials and come from setting restrictions on the support.…”

“…The associated partial Bell polynomials (2.3) and (2.4) are represented by another partial Bell polynomial or as a linear combination of other partial Bell polynomials [10]. For later discussion, we present the following fact, which was not presented in [10].…”

“…The unique polynomial solutions of this A-hypergeometric system are constant multiples of the A-hypergeometric polynomial. In particular, for the A-hypergeometric system associated with a special monomial curve, called the rational normal curve, the Ahypergeometric polynomial is a constant multiple of the associated partial Bell polynomial, which was recently defined by the author [10].…”