Limit Theorems in Probability Theory

概率论中的极限定理

• 批准号: 0070382
• 负责人: Evarist Gine
• 金额: $ 7.5万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070382&HistoricalAwards=false
• 关键词: Limit; Theorems; Probability; Theory

0070382Gine Work is planned on several topics from asymptotic theory in Probability and Statistics. A main thrust of the research aims at deepening our understanding of canonical $U$-statistics and $U$-processes by investigating exponential and moment inequalities (what are the true analogues for $U$-statistics of the Rosenthal-Pinelis and Bernstein's inequalities? is there a uniform Bernstein, or uniform Prohorov inequality such as the recent inequality of Talagrand for collections of sums of independent random variables?) and limit theorems, particularly the law of the iterated logarithm. These results may be obtained for generalized $U$-statistics, including multilinear forms in independent random variables. Applications of these topics in Statistics, particularly censored data, will also be pursued. A second object of study are selfnormalized sums of independent random variables, particularly in connection with the bootstrap and with the Student t-statistic. Finally, the P.I. is also interested in exploring the application of the modern theory of empirical processes and its techniques in different areas such as asymptotics of the fluctuations of the occupation measure process for multiple particle systems, and estimation and testing based on different functionals of the empirical process. The empirical measure is shorthand for the description of a series of data points. Sums of independent random variables and empirical processes can be thought of as single integrals of functions of one variable with respect to this measure, and $U$-statistics and $U$-processes, as multiple integrals with respect to the empirical measure of functions of several variables. First order asymptotic statistics is often based on limit theorems for sums of independent random variables and processes, but more refined second order properties require limit theory for $U$-statistics and processes (in a way, in analogy with the use of higher order derivatives versus only the first derivative when studying functions in Calculus). Although $U$-statistics were introduced in the forties, their asymptotic theory has not been close to reaching its final form until recently, in part due to previous efforts by this P.I. and collaborators; the proposed research aims at completing this chapter of Classical Probability for $U$-statistics, and at advancing the theory of $U$-processes, by obtaining best possible distributional and moment inequalities and laws of the iterated logarithm. This research will also include applications in survival analysis. In another direction, it is accepted wisdom that normalizing sums of independent random variables by certain quantities that depend on themselves rather than numerical constants improves the convergence properties (in particular, then, statistical procedures based on such selfnormalized quantities may have good properties, the leading and oldest example of this being the famous Student t-statistic and test). But this must be shown at each instance. The P.I. would like to study some questions related to selfnormalized sums, particularly in connection with the bootstrap. Empirical process theory vigorously developed during the last two decades (with substantial contributions by this P.I.) and, since then, its impact on different fields of stochastics has not ceased to increase (in classical asymptotic statistics, information theory, neural networks, machine learning, model selection, statistical mechanics, etc.), and the P.I. would like to continue applying it to different statistics and probability problems of current interest.

工作计划在概率论和统计学中的渐近理论的几个主题上。该研究的主要目的是通过调查指数和矩不等式（Rosenthal-Pinelis和Bernstein的不等式的U -统计的真正类似物是什么？）来加深我们对标准U -统计和U -过程的理解。是否有一个统一的Bernstein不等式，或者统一的Prohorov不等式，比如最近的Talagrand不等式，用于独立随机变量和的集合？)和极限定理，特别是迭代对数定律。这些结果可用于广义的$U$统计量，包括独立随机变量中的多元线性形式。还将探讨这些专题在统计学中的应用，特别是审查后的数据。第二个研究对象是独立随机变量的自归一化和，特别是与自举和学生t统计量有关。最后，P.I.也有兴趣探索经验过程的现代理论及其技术在不同领域的应用，如多粒子系统的职业测量过程波动的渐近性，以及基于经验过程的不同泛函的估计和测试。经验测度是对一系列数据点的描述的简写。独立随机变量和经验过程的和可以被认为是单变量函数相对于该测度的单积分，而U -统计和U -过程可以被认为是相对于多变量函数的经验测度的多重积分。一阶渐近统计通常基于独立随机变量和过程的和的极限定理，但更精细的二阶性质需要$U$统计和过程的极限理论（在某种程度上，类似于在微积分中学习函数时使用高阶导数而不是只使用一阶导数）。虽然$U$统计在40年代被引入，但直到最近，他们的渐近理论才接近最终形式，部分原因是由于该pi和合作者先前的努力；提出的研究旨在完成这一章的经典概率$ $-统计，并在推进$ $-过程的理论，通过获得最佳可能的分布和矩不等式和规律的迭代对数。这项研究还将包括在生存分析中的应用。在另一个方向上，人们公认的智慧是，通过依赖于自身而不是数值常数的某些量来规范化独立随机变量的和，可以提高收敛性（特别是，基于这种自规范化量的统计过程可能具有良好的性质，这方面的主要和最古老的例子是著名的学生t统计和检验）。但这必须在每个实例中显示出来。P.I.想研究一些与自规格化和有关的问题，特别是与自举有关的问题。经验过程理论在过去二十年中得到了大力发展（该P.I.做出了重大贡献），从那时起，它对随机学不同领域的影响一直在增加（在经典渐近统计、信息论、神经网络、机器学习、模型选择、统计力学等），而P.I.希望继续将其应用于当前感兴趣的不同统计和概率问题。