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

点击翻译按钮获取中文摘要

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.