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.

建议研究一些概率问题。一类问题涉及复杂对象的中心限制定理，例如最近的邻居统计数据和大尺寸随机矩阵特征值的线性统计。 PI发明了一种新方法，该方法是通过利用正常近似和度量浓度（概率理论的两个不同分支）之间未知的连接而起作用的。第二类问题涉及Lebesgue测量对欧几里得空间和歧管中离散点过程的公平分配。最后，第三次调查追求一种将相互作用的布朗颗粒分析与凸多图形的几何形状联系起来的方法。该项目的关键重点是中心极限定理。众所周知，CLT是概率理论的创始支柱之一，可以说是在应用科学中使用的最广泛使用的工具，在从统计学到生物信息学，计算机科学再到经济学的田野中找到了日常应用。尽管众所周知，但仍然有许多未解决的问题。实际上，PI对中央限制理论定理的调查是由彼得·比克尔（Peter Bickel）提出的，彼得·比克尔（Peter Bickel）是一位在高维数据分析领域工作的著名伯克利科学家。 PI现在有了一种新技术来证明CLT不仅解决了空旷的问题，而且屈服并承诺会产生更多的收益。