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

建议研究概率论中的几个问题。一类问题涉及复杂对象的中心极限定理，如最近邻统计量和大维随机矩阵特征值的线性统计量。PI发明了一种新的方法，它利用了正态近似和测量集中之间迄今未知的联系，这是概率论的两个不同分支。第二类问题涉及到欧氏空间和流形中离散点过程的勒贝格测度的公平分配。最后，第三条线探索一种将相互作用的布朗粒子的分析与凸多面体的几何联系起来的方法。该项目的主要焦点是中心极限定理。众所周知，CLT是概率论的奠基支柱之一，可以说是其在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的各个领域都有日常应用。尽管已经知道了很多，但仍有许多悬而未决的问题。事实上，PI对中心极限定理理论的调查是由伯克利分校在高维数据分析领域工作的著名科学家Peter Bickel提出的一个公开问题发起的。PI现在有了一种新的技术来证明CLT，它不仅解决了这个悬而未决的问题，而且已经屈服，并承诺产生更多。

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.