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

泊松-狄利克雷分布：随机动力学、准不变性和渐近性

• 批准号: 155745-2011
• 负责人: Feng, Shui
• 金额: $ 1.17万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=476099
• 关键词: Poisson; Dirichlet; distribution; stochastic; dynamics

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.

Poisson-Dirichlet分布是一种两参数概率分布，由于其在广泛的学科中的应用，包括但不限于贝叶斯统计，生物学，结合，生态学，经济学，财务，语言学，语言学，数量理论，物理学和人口遗传学，因此变得越来越重要。 与分布相关的研究已经在世界各地的许多研究中心进行，研究活动主要集中在随机进化模型，一般的随机组合结构，渐近行为和各种应用上。拟议的研究是该大型国际计划的一部分，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主要是无限维的随机过程。此处的目的是找到具有泊松二甲基分布或其他相关分布的平衡分布的随机过程。 这些过程的发现将提供有关潜在的进化机制以及人口有限时间和长期状态之间的关系的深刻见解。它还将导致统计推断中的新算法。 这项研究的另一个组成部分涉及与分布相关的各种统计的渐近行为。已知Poisson-Dirichlet分布在参数域的不同部分具有截然不同的行为。该领域的三个主要区域是高斯，中间和非常非高斯。研究的这一部分旨在研究不同地区之间的详细过渡，并更好地了解不同进化力的影响和相互作用。