Limit Theorems for Various Dependence Structures with Applications to Statistics and Particle Systems

本研究主要研究随机变量的相依结构和相关的大样本分布。预计结果将包括单变量和多变量正态近似、中心极限定理和相依和分布的级数展开。这些结果将扩展假设独立的经典理论，并允许在统计学、组合学和计算机科学中有许多新的应用，在这些领域中，人们可以处理因变量的和。斯坦的方法在提案中发挥了重要作用，但其他方法也是特色。对收敛速度进行了仔细的研究。该方案描述了博弈论、大样本理论、统计力学中的粒子系统等的应用。大粒子系统、大型统计调查、大型计算系统等的概率行为的研究通常是通过考虑随着系统规模的增长而限制，并假设粒子或调查意见等的独立性来进行的。然而，对于许多系统或抽样方案，随机结果之间的相关性是不可忽视的。这项研究集中于研究存在依赖的大系统或样本的方法，以及在各种依赖结构下的研究方法。研究了有限系统逼近其极限行为的速率，从而允许将结果应用于给定有限大小的系统。

9803625RinottThis research focuses on dependence structures of random variables and associated large sample distributions. Results are expected to include univariate and multivariate normal approximations, Central Limit Theorems and series expansions of distributions for dependent summands. Such results will extend the classical theory, which assumes independence, and allow many new applications in statistics, combinatorics and computer science, where one deals with sums and counts of dependent variables. Stein's methods play an important role in the proposal, but other methods are featured as well. A careful study of convergence rates is proposed. Applications to game theory, large sample theory, particle systems in statistical mechanics and others, are described in the proposal.The probabilistic behavior of large systems of particles, large statistical surveys, large computational systems, etc., is often studied by considering limits as the size of the system grows, and assuming independence of the particles, or opinions surveyed etc. However, for many systems or sampling schemes, the dependence between the random outcomes cannot be ignored. This research concentrates on methods of studying large systems or samples in the presence of dependence, and under various dependence structures. The rate at which finite systems approach their limiting behavior is studied, thus allowing applications of the results to systems of a given finite size.