Small Value Theory in Probability

概率中的小值理论

• 批准号: 0805929
• 负责人: Wenbo Li
• 金额: $ 16万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805929&HistoricalAwards=false
• 关键词: Small; Value; Theory; Probability

Two fundamental phenomenons in probability theory are typical behaviors such as expected values, laws of large numbers and central limit theorems, and rare events such as extremely big or small values. This research centers on developing methods and theory for the study of both typical behaviors and rare events of the type that positive random quantities take smaller values. The major objective is to extend the understanding of five related areas and build a general small value theory based on systematic study of various techniques and applications. The isoperimetric type Gaussian inequalities provide comparisons between dependent (complected) structure and independent (simpler) one which becomes an equality in certain (possibility limiting) cases. They have been used as basic tools in various problems and played a crucial role in deeper understanding of random phenomenon. The recent development of several new techniques for Gaussian and closely related random processes broadened our understanding of small deviation probabilities and their connections with related topics of probability such as Gaussian random matrices, non-intersection exponents and random assignments.In turn, it suggests many further questions connected to applications in probability theory and geometric functional analysis. The very successful applications to lower tail probabilities, zeros of random functions and the first exit times will be expanded to a detailed study of Brownian pursuit models and Gaussian chaos.This research has a broader impact on diverse areas of probability, which is both a fundamental way of viewing the world and a core mathematical discipline. The theory of Gaussian (bell curve) processes is of fundamental importance in probability and statistics. Its development is centered on applications of the existing methods to a variety of fields and new techniques and problems motivated by current interests of advancing knowledge. The proposed research is a key step in the investigator's long term research plan of systematically developing new Gaussian methods geared for applications to closely related random processes. This research benefits both undergraduate and graduate education and research. Many open problems and results from the proposed study can be used as students course projects. It can stimulate the interests of students in leaning and studying probability theory. This research should improve our understanding of important random events and provide basic tools for the study of our random environment.

概率论的两个基本现象是期望值、大数定律、中心极限定理等典型行为和极小值等极小值等罕见事件。本研究的重点是发展研究典型行为和罕见事件的方法和理论，即正随机量取较小值的类型。主要目标是扩展对五个相关领域的理解，并在系统研究各种技术和应用的基础上建立一个通用的小价值理论。等周型高斯不等式提供了依赖（完整）结构和独立（更简单）结构之间的比较，这在某些（可能性限制）情况下成为一个等式。它们被用作解决各种问题的基本工具，在深入理解随机现象方面发挥了至关重要的作用。最近发展的几种高斯和密切相关的随机过程的新技术拓宽了我们对小偏差概率的理解，以及它们与概率的相关主题，如高斯随机矩阵、非相交指数和随机赋值的联系。反过来，它提出了许多与概率论和几何泛函分析应用有关的进一步问题。对低尾概率、随机函数的零和首次退出时间的非常成功的应用将扩展到对布朗追逐模型和高斯混沌的详细研究。这项研究对概率的各个领域有着更广泛的影响，这既是一种观察世界的基本方式，也是一门核心的数学学科。高斯（钟形曲线）过程理论在概率论和统计学中具有重要的基础意义。它的发展集中于现有方法在各种领域的应用，以及由当前对先进知识的兴趣所激发的新技术和新问题。该研究是研究者长期研究计划的关键一步，该计划旨在系统地开发新的高斯方法，以应用于密切相关的随机过程。这项研究有利于本科和研究生的教育和研究。许多未解决的问题和建议研究的结果可以用作学生的课程项目。它能激发学生学习和研究概率论的兴趣。这项研究将提高我们对重要随机事件的理解，并为研究我们的随机环境提供基本工具。