Exact inequalities and limit theorems for Rademacher and self-normalized sums, and related statistics

该项目的主要目标如下：*证明了Rademacher-Gaussian尾部比较中关于最佳常数因子的长期猜想。*证明另一个长期存在的猜想，关于Rademacher尾被高斯尾渐近支配。再考虑一下“不对称”的情况。*将Shao等人关于独立随机变量自归一化和的大偏差概率的鞍点近似的结果推广到中等偏差的情况。*获得Pearson积矩样本相关系数的极限定理，包括berry - esseen型边界和cramer型大偏差渐近性，以及一些类似的更一般的统计量。因此，研究者的目标是解决长期和困难的问题，概率论和数理统计。它们中的前两个涉及Rademacher和这类经典和基本对象的一些最重要的性质，Rademacher和的分布充当任意独立对称随机变量的和（和自归一化和）分布集合的极值点的作用。还将考虑“不对称”情况的扩展。密切相关的是项目的其他主要目标，关于自规格化和的极限定理（或等价地，对于学生的统计）。其主要影响将是对概率论和数理统计中一些最基本对象的重要性质有更好的理解。该项目的成功完成还将导致对诸如学生检验和皮尔逊相关检验等统计学经典对象的新颖和重要应用，这是在科学和工程中最广泛使用的极少数假设检验中的一些。虽然有很大的困难需要克服，但鉴于研究者已经取得的一些进展，以及他在概率和统计各个领域的相当独特的专业知识，以及他在识别和解决困难和长期存在的问题以及在广泛和高度多样化的领域有效工作的能力，这些目标的实现似乎是可以实现的，包括机械工程，生物学，运筹学和组合学，几何学和物理学。将努力传播研究结果，不仅通过在发行量很大的期刊上发表，而且还通过新闻网络传播（合众国际社和其他新闻机构已经向全世界广播了关于研究者在进化模型和埃菲尔铁塔形状模型方面的工作）。一些研究生将参与该项目；将努力从代表性不足的少数民族中招聘。

The main objectives of the project are as follows: * Prove the longstanding conjecture on the best constant factor in the Rademacher-Gaussian tail comparison. * Prove another longstanding conjecture, on the asymptotic domination of the Rademacher tail by the Gaussian one. * Consider also the ``asymmetric'' case. * Extend to the case of moderate deviations the result due to Shao et al. on the saddle-point approximation to large-deviation probabilities of a self-normalized sum of independent random variables. * Obtain limit theorems, including Berry-Esseen-type bounds and Cramer-type large-deviation asymptotics, for Pearson's product-moment sample correlation coefficient and a number of similar and more general statistics. Thus, the investigator aims to solve longstanding and difficult problems of probability theory and mathematical statistics. The first two of them concern some of the most important properties of such a classical and fundamental object as the Rademacher sums, whose distributions play the role of the extreme points of the set of the distributions of sums (and self-normalized sums) of any independent symmetric random variables. Extensions to the ``asymmetric'' case will also be considered. Closely related are other main objectives of the project, concerning limit theorems for self-normalized sums (or, equivalently, for Student's statistic). The main impact will be in significantly better understanding of important properties of some of the most fundamental objects in probability theory and mathematical statistics. The successful completion of the project will also result in novel and important applications to such classical objects in statistics as Student's test and Pearson's correlation test, which are some of the very few hypotheses tests used most broadly in sciences and engineering. While there are great difficulties to overcome, it appears that the attainment of these objectives is within reach, given a number of advances already made by the investigator and his rather unique expertise in various areas of probability and statistics, as well as his demonstrated abilities to identify and solve difficult and longstanding problems and also to work effectively in a wide and highly diverse range of fields, including mechanical engineering, biology, operations research and combinatorics, and geometry and physics. Efforts will be made to disseminate results, not only via publication in wide-circulation journals, but also via news networks (stories on the investigator's work on evolution modeling and the Eiffel tower shape modeling have already been broadcast around the world by the United Press International and other news agencies). A number of graduate students will be involved into the project; efforts will be made to recruit from underrepresented minorities.