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

Rademacher 和自归一化和的精确不等式和极限定理以及相关统计

• 批准号: 0805946
• 负责人: Iosif Pinelis
• 金额: $ 15万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805946&HistoricalAwards=false
• 关键词: Exact; inequalities; limit; theorems; Rademacher

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.

该项目的主要目标如下： *证明了Rademacher-Gaussian尾巴比较中最佳恒定因素的长期猜想。 *证明了另一个长期以来的猜想，在高斯尾巴对rademacher尾巴的渐近统治上。 *还考虑``不对称''案例。 *扩展到中等偏差的情况，其结果是Shao等人引起的。关于独立随机变量的自相应总和的大差异概率的鞍点近似。 *获取限制定理，包括浆果 - 式型边界和cramer型大型渐近学，用于Pearson的产品量像样品相关系数以及许多相似且更一般的统计数据。因此，研究者旨在解决概率理论和数学统计的长期存在和困难的问题。他们中的前两个涉及这样的经典和基本对象的一些最重要的属性，例如Rademacher总和，其分布发挥了任何独立对称随机变量的总和（和自分配总和）的极端点的作用。还将考虑``非对称''案件的扩展。密切相关的是该项目的其他主要目标，涉及限制自称为单位的定理（或等效地，对于学生的统计数据）。主要的影响将是对概率理论和数学统计中一些最基本对象的重要特性的重要理解。该项目的成功完成还将在学生的测试和Pearson的相关测试等统计数据中为此类经典对象提供新颖和重要的应用，这是科学和工程中最广泛使用的极少数假设测试中的一些。虽然有很大的困难要克服，但似乎达到了这些目标的实现，鉴于研究者已经取得了许多进步，以及他在概率和统计数据的各个领域中已经取得了许多进步，以及他在各个概率和统计领域都取得了独特的专业知识，以及他在各种领域和高度多样化的机构工具和范围内的范围和多样化的范围和多样化的范围和高度的机构研究和综合处理，以及在各种多样化的领域中识别和解决了长期的问题，以及经常性和综合。不仅通过广泛期刊的出版物，而且还通过新闻网络（研究人员关于进化建模的故事和Eiffel Tower Shape建模的故事已经在世界范围内广播着联合国出版社国际新闻国际和其他新闻机构的世界播出）。许多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募。