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.

0070382GINE工作计划在概率和统计学方面的渐近理论中的几个主题。 这项研究的主要力量旨在通过调查指数和瞬间不平等来加深我们对规范$ u $ $ $ $的统计和$ u $ - 过程（$ u $ $ u $ statigation的真正类似物是罗斯塔尔·皮尼利斯（Rosenthal-Pinelis）和伯恩斯坦（Bernstein）的不平等现象的真正类似物？独立随机变量的总和？）和限制定理，特别是迭代对数定律。 可以为广义$ u $统计数据获得这些结果，包括独立随机变量中的多线性形式。 这些主题在统计数据中的应用，尤其是经过审查的数据，也将被追求。 研究的第二个对象是独立随机变量的自称总和，尤其是与引导程序以及与学生t统计症相关。 最后，P.I.还有兴趣探索现代经验过程理论及其在不同领域的技术的应用，例如多种粒子系统的占用度量过程的渐近学，以及基于经验过程的不同功能的估计和测试。经验度量是描述一系列数据点的速记。 独立的随机变量和经验过程的总和可以被视为相对于此度量的一个变量的函数的单个积分，以及$ u $统计信息和$ u $ - 过程，作为多个积分相对于多个变量函数的经验度量的多个积分。 一阶渐近统计通常是基于独立随机变量和过程总和的极限定理，但是更精致的二阶属性需要$ u $统计量和过程的极限理论（以某种方式类似于使用高阶衍生物，而仅在计算功能中仅使用第一个衍生功能）。 尽管四十年代引入了$ U $统计量，但直到最近，由于这一P.I的先前努力，它们的渐近理论直到最近才达到其最终形式。和合作者；拟议的研究旨在通过获得迭代对数的最佳分配和时刻不平等和法律来完成$ u $统计学的经典概率的这一章，并促进$ u $ - 过程的理论。 这项研究还将包括生存分析中的应用。在另一个方向上，人们接受的智慧是，将独立随机变量的标准化归一定量的依赖于自己而不是数值常数提高了收敛性能（尤其是，特别是，基于此类自降低数量的统计程序可能具有良好的属性，这可能具有良好的属性，这是著名的学生T-Statistic t-Statistist和Test的最大典型示例）。 但这必须在每个实例中显示。 P.I.想研究一些与自称的总和有关的问题，尤其是与引导程序有关。 在过去的二十年中，经验过程理论在过去的二十年中积极发展（该P.I.具有实质性的贡献），从那时起，其对随机不同领域的影响尚未停止增加（在经典的渐近统计，信息理论，神经网络，机器学习，机器学习，模型选择，统计机制等）等。希望继续将其应用于当前关注的不同统计数据和概率问题。