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

该项目的主要目标如下：* 证明关于Rademacher-Gaussian尾比较中最佳常数因子的长期猜想。* 证明另一个长期存在的猜想，关于高斯尾对拉德马赫尾的渐近支配。* 再看“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。* 获得极限定理，包括Berry-Esseen型界和Cramer型大偏差渐近，皮尔逊积矩样本相关系数和一些类似的和更一般的统计量。因此，研究者旨在解决概率论和数理统计的长期和困难的问题。前两个问题涉及到Rademacher和这样一个经典的基本对象的一些最重要的性质，Rademacher和的分布起着任何独立的和（和自正规化和）的分布集的极值点的作用。 对称随机变量还将考虑对“不对称”情况的扩展。密切相关的是其他主要目标的项目，关于极限定理的自我规范化的总和（或，相当于学生的统计）。主要的影响将是在显着更好地了解一些最基本的对象在概率论和数理统计的重要属性。该项目的成功完成也将导致在新颖的和重要的应用，如学生的测试和皮尔逊的相关性测试，这是一些非常少的假设测试，最广泛地使用在科学和工程统计等经典对象。虽然有很大的困难要克服，但这些目标似乎是可以实现的，因为调查员已经取得了一些进展，他在概率和统计学的各个领域具有相当独特的专门知识，他有能力查明和解决长期存在的困难问题，并在广泛和高度多样化的领域有效地开展工作，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，不仅通过在发行量大的期刊上发表，而且通过新闻网络传播（合众国际社和其他新闻机构已经向全世界广播了研究人员关于进化建模和埃菲尔铁塔形状建模的报道）。一些研究生将参与该项目;将努力从代表性不足的少数群体中征聘人员。

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.