The Poisson-Dirichlet distribution: stochastic dynamics, quasi-invariant, and asymptotics

泊松-狄利克雷分布是一种双参数概率分布，由于其在包括但不限于贝叶斯统计、生物学、组合学、生态学、经济学、金融学、语言学、数论、物理学和群体遗传学等广泛学科中的应用而变得越来越重要。世界上许多研究中心都开展了与分布相关的研究，研究活动主要集中在随机进化模型、一般随机组合结构、渐近行为和各种应用。拟议的研究是这个大型国际项目的一部分，将处理分布的两个主要方面：随机进化模型和渐近行为。该程序中的随机进化模型多为无限维随机过程。这里的目标是找到具有泊松-狄利克雷分布或其他相关分布给出的平衡分布的随机过程。这些过程的发现将对潜在的进化机制以及种群有限时间和长期状态之间的关系提供深刻的见解。它还将导致统计推断中的新算法。本研究的另一个组成部分是与分布相关的各种统计的渐近行为。已知泊松-狄利克雷分布在参数域的不同部分具有显著不同的行为。该领域的三个主要区域是高斯、中间和非常非高斯。这部分研究的目的是研究不同区域之间的详细转变，更好地理解不同进化力量的影响和相互作用。

The Poisson-Dirichlet distribution is a two-parameter probability distribution that has become increasingly important due to its applications in a wide range of disciplines including but not limited to Bayesian statistics, biology, combinatorics, ecology, economics, finance, linguistics, number theory, physics and population genetics. The research associated the distribution has been carried out in many research centres around the world and the research activities focus mainly on the stochastic evolutionary models, general random combinatorial structures, asymptotic behaviour, and various applications. The proposed research is part of this large international program and will deal with two main aspects of the distribution: the stochastic evolutionary models and asymptotic behaviour. The stochastic evolutionary models in this program are mostly infinite-dimensional stochastic processes. The objective here is to find stochastic processes that have equilibrium distributions given by either the Poisson-Dirichlet distribution or other related distributions. The discovery of these processes will provide deep insights on the underlying evolutionary mechanism and the relationship between finite-time and long-time states of the population. It will also lead to new algorithms in statistical inference. Another component of this research is concerned with the asymptotic behaviours of various statistics related to the distribution. The Poisson-Dirichlet distribution is known to have dramatically different behaviours at different parts of the parameter domain. Three main regions of the domain are the Gaussian, intermediate, and very non-Gaussian. This part of the research aims to study the detailed transitions between different regions and to better understand the impact and interplays of different evolutionary forces. The techniques and methods developed in this program will undoubtedly enrich our knowledge in probability theory and stochastic processes. The results obtained will be beneficial for a wide range of subjects. The trainees participating in this research program will be very well prepared for academic and industrial careers.