Normal Approximation, Fair Allocations, Interacting Brownian Particles, and Applications

正态近似、公平分配、相互作用的布朗粒子和应用

• 批准号: 0707054
• 负责人: Sourav Chatterjee
• 金额: $ 13万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707054&HistoricalAwards=false
• 关键词: Normal; Approximation; Fair; Allocations; Interacting

It is proposed to study several problems in Probability. One class of problems concerns central limit theorems for complex objects like nearest neighbor statistics and linear statistics of eigenvalues of large dimensional random matrices. The PI has invented a new method that works by exploiting a hitherto unknown connection between normal approximation and concentration of measure, two different branches of probability theory. A second class of problems involves fair allocations of Lebesgue measure to discrete point processes in Euclidean spaces and manifolds. Finally, a third line of investigation pursues a method of connecting the analysis of interacting Brownian particles with the geometry of convex polytopes.The key focus of the project is on Central Limit Theorems. CLT's, as they are popularly known, are one of the founding pillars of Probability Theory and arguably its most widely used tool in the applied sciences, finding everyday applications in fields ranging from Statistics to Bio-informatics, Computer Science to Economics. Although much is known, there are still many unsolved questions. In fact, the PI's investigation into the theory of Central Limit Theorems was initiated by an open question raised by Peter Bickel, an eminent Berkeley scientist working in the area of high dimensional data analysis. The PI now has a new technique for proving CLT's that has not only solved the open question, but has yielded and promises to yield much more.

建议研究概率中的几个问题。一类问题涉及复杂对象的中心极限定理，例如最近邻统计和大维随机矩阵特征值的线性统计。 PI 发明了一种新方法，其工作原理是利用正态近似和测量集中（概率论的两个不同分支）之间迄今为止未知的联系。第二类问题涉及将勒贝格测度公平分配到欧几里得空间和流形中的离散点过程。最后，第三条研究路线寻求一种将相互作用的布朗粒子分析与凸多面体几何联系起来的方法。该项目的重点是中心极限定理。众所周知，CLT 是概率论的奠基支柱之一，并且可以说是应用科学中使用最广泛的工具，在从统计学到生物信息学、计算机科学到经济学等领域都有日常应用。尽管已知的很多，但仍有许多未解决的问题。事实上，PI 对中心极限定理理论的研究是由 Peter Bickel 提出的一个开放性问题发起的，Peter Bickel 是一位在高维数据分析领域工作的伯克利杰出科学家。 PI 现在拥有一种证明 CLT 的新技术，该技术不仅解决了悬而未决的问题，而且已经产生并有望产生更多成果。