A hydrodynamic limit and fluctuations for a Chinese restaurant-like process

中国餐馆过程中小块的比例极限

Scaling limit for small blocks in the Chinese restaurant process

本文研究中国餐馆过程中小块组成的随机点测度，证明其比例极限为递增维锥上非齐次泊松测度的射影极限，从而推导Skorokhod拓扑下的经典及函数极限定理。

可交换Gibbs分拆的广义中餐厅构造及相关结果

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。

Diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains

We construct a two-parameter family of Feller diffusions on the set of open subsets of $(0,1)$ that arise as diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains. The diffusions we construct are natural ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. Recently, there has been significant interest in ordered analogues of the diffusions Petrov constructed. Existing methods for constructing such processes have been based on pathwise methods using marked Lévy processes and an outstanding conj…

A Ray-Knight representation of up-down Chinese restaurants

We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (2009). We relate such up-down CRPs to the splitting trees of Lambert (2010) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray-Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establi…

The leftmost column of ordered Chinese Restaurant Process up-down chains: intertwining and convergence

Recently there has been significant interest in constructing ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. One method for constructing these processes goes through taking an appropriate diffusive limit of Markov chains on integer compositions called ordered Chinese Restaurant Process up-down chains. The resulting processes are diffusions whose state space is the set of open subsets of the open unit interval. In this paper we begin to study nontrivial aspects of the order structure of these diffusions. In particular,…

Concentration in the Generalized Chinese Restaurant Process

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…