Topics in Small Value Theory of Probability

小值概率论专题

• 批准号: 1106938
• 负责人: Felix Lazebnik
• 金额: $ 30万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106938&HistoricalAwards=false
• 关键词: Topics; Small; Value; Theory; Probability

This research project considers the development of various techniques in the study of small value theory: Both typical behaviors and rare events of the type that positive random quantities take smaller values.The typical behavior deals with the expectation of the minimum over a family of non-negative random variables as the size of the index set grows larger.Topics in this direction includeminimum length spanning tree indexed by spanning trees, comparison inequalities for minimum of the absolute value of Gaussian vectors, random assignment type problems indexed by permutations, and the first passage percolation indexed by paths.The rare events of small value type deal with decay probabilities of positive random quantities taking smaller values than typical ones.Topics in this direction include small ball probabilities, lower tail behaviors and level crossing probabilities.The very successful applications of tools developed by the proposer over the last few years will be expanded to a detailed study of various isoperimetric type Gaussian inequalities, tensored random fields, Gaussian chaos, Brownian eddy models, permanental processes, and positivity exponents for random polynomials.The major objective is to build and extend a general small value theory based on systematic study of various methods and diverse applications.Two fundamental phenomena 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 proposal aims to deepen our understanding of small value phenomena for positive random quantities by developing new techniques in the study of relevant physical and biological random models.This research benefits both undergraduate and graduate education and research. Many open problems and results from the proposed study will be used as students course projects.This research should improve our understanding of extremal random events and provide basic tools for the study of our random environment.

本研究项目考虑了小值理论研究中各种技术的发展：正随机量取较小值的典型行为和罕见事件。典型行为涉及随着索引集的大小变大，非负随机变量族的最小值的期望。这方面的主题包括由生成树索引的最小长度生成树，高斯向量绝对值最小值的比较不等式，排列索引的随机分配类型问题，以及路径索引的第一次通过渗流。小值类型的罕见事件处理比典型事件取更小值的正随机量的衰减概率。这个方向的主题包括小球概率，下尾行为和水平交叉概率。在过去几年中，提议者开发的工具的非常成功的应用将扩展到各种等周型高斯不等式，张量随机场，高斯混沌，布朗涡模型，永久过程，随机多项式的正指数。主要目标是在系统研究各种方法和各种应用的基础上建立和扩展一般的小值理论。概率论中的两个基本现象是典型的行为，如期望值，大数定律和中心极限定理，和罕见事件，如极大值或极小值。这项建议旨在通过发展相关物理和生物随机模型研究的新技术，加深我们对正随机量小值现象的理解。这项研究对本科和研究生教育和研究。许多开放的问题和结果，从拟议的研究将被用作学生的课程projects.This研究应该提高我们的极端随机事件的理解，并为我们的随机环境的研究提供基本的工具。