The Pitman sampling formula has been intensively studied as a distribution of random partitions. One of the objects of interest is the length $K (= K_{n,θ,α})$ of a random partition that follows the Pitman sampling formula, where $n\in\mathbb{N}$, $α\in(0,\infty)$ and $θ> -α$ are parameters. This paper presents asymptotic evaluations of its $r$-th moment $\mathsf{E}[K^r]$ ($r=1,2,\ldots$) under two asymptotic regimes. In particular, the goals of this study are to provide a finer approximate evaluation of $\mathsf{E}[K^r]$ as $n\to\infty$ than has previously been developed and to provide an approximate evaluation of $\mathsf{E}[K^r]$ as the parameters $n$ and $θ$ simultaneously tend to infinity with $θ/n \to 0$. The results presented in this paper will provide a more accurate understanding of the asymptotic behavior of $K$.
We introduce a continuous-time Markov chain describing dynamic allelic partitions which extends the branching process construction of the Pitman sampling formula in Pitman (2006) and the birth-and-death process with immigration studied in Karlin and McGregor (1967), in turn related to the celebrated Ewens sampling formula. A biological basis for the scheme is provided in terms of a population of individuals grouped into families, that evolves according to a sequence of births, deaths and immigrations. We investigate the asymptotic behaviour of the chain and show that, as opposed to the birth-a…
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, …
Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results
借助于可交换随机分拆的序列构造（Pitman，2006），并利用关于广义Stirling数的一些已知事实，我们得到了$α$型可交换Gibbs分拆的广义中餐厅过程构造（Gnedin and Pitman，2006）.我们的建设代表了自然的理论概率框架，其中嵌入一些最近的结果贝叶斯非参数处理的估计问题，在遗传实验吉布斯，物种抽样，模型先验。
The Ewens-Pitman sampling model (EP-SM) is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, which is index by real parameters $α$ and $θ$ such that either $α\in[0,1)$ and $θ>-α$, or $α<0$ and $θ=-mα$ for some $m\in\mathbb{N}$. For $α=0$ the EP-SM reduces to the celebrated 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 generalization of the LS-CPSM, which is referred to as the negative Binomial compound Poisson sampling m…
Pitman(2003)(and subsequently Gnedin and Pitman (2006) showed that a large class of random partitions of the integers derived from a stable subordinator of index $α\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson-Dirichlet distribution, $\mathrm{PD}(α,θ)$, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(α,θ).$ Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it re…
This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability gener…
The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers $\{1,\dots,n\}$. This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like $n^α$ with $α>0$. We prove a non-asymptotic concentration result for the number of parts of size $k=o(n^{α/(2α+4)}/(\log n)^{1/(2+α)})$. In particular, we show that these random variables concentrate around $c_{k}\,V_*\,n^α$ where $V_*\,n^α$ is the asymptot…
