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

提出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，如最近邻统计和大维随机矩阵特征值的线性统计。PI发明了一种新方法，该方法利用了概率理论的两个不同分支正态近似和测量浓度之间迄今为止未知的联系。第二类问题涉及欧几里得空间和流形中离散点过程的勒贝格测度的公平分配。最后，第三条研究路线追求将相互作用布朗粒子的分析与凸多面体的几何形状联系起来的方法。这个项目的重点是中心极限定理。众所周知，CLT是概率论的基础支柱之一，也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，从计算机科学到经济学等领域都有日常应用。虽然我们知道很多，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提出的一个公开问题发起的，Peter Bickel是一位杰出的伯克利科学家，研究高维数据分析领域。PI现在有了一种证明CLT的新技术，它不仅解决了这个悬而未决的问题，而且已经产生并有望产生更多的结果。

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.