General Limit Theorems in Probability with Applications to Statistics

概率的一般极限定理及其在统计中的应用

• 批准号: RGPIN-2014-05428
• 负责人: Li, Deli
• 金额: $ 1.02万
• 依托单位: Lakehead University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=657430
• 关键词: General; Limit; Theorems; Probability; Applications

My research proposal is devoted to general limit theorems in probability and asymptotic statistics and in their applications to a wide variety of problems.**We live in a random world. Probability theory is the branch of mathematics concerned with analysis of random phenomena. Limit theory lies at the heart of probability and statistics and plays a central stage in almost every branch of science or social science including weather forecasting, psephology, etc. Results such as the law of large numbers, the central limit theorem, and the law of the iterated logarithm for independent random variables have given shape to modern probability theory. They have been extended and generalized in many directions, among others, to more general random processes and random measures, and they have become the bases of asymptotic statistics. Asymptotic statistics or large sample theory, is a generic framework for the assessment of properties of estimators and statistical tests. Within this framework, it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as the sample size n tends to infinity.**The first focus of this research proposal relates to my my long-standing research interest in almost sure and weak convergence of random processes, especially in the law of the iterated logarithm, the laws of large numbers, central limit theorems, probabilities of large and moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach space-valued random processes. The goal is to study refinements of the classical limit results and to develop some new methods for proving almost sure and weak convergence of random processes and to continue my previous research work, i.e., to use modern random process techniques in probability and to develop some new probability inequalities for random processes in order to find conditions under which almost sure and weak convergence holds for random processes and to investigate statistical applications of such convergence.**A second focus will be on investigating the asymptotic behavior in statistical applications pertaining to hierarchical models, L-statistics, U-statistics, resampling methods, and high dimensional data analysis problems such as the largest entry of a sample correlation matrix, estimation of conditional density and mode with truncated and censored data, etc. For example, motivated by a statistical hypothesis testing problem, asymptotic behavior of the largest entry of a sample correlation matrix has been studied extensively in recent years including my three refereed journal articles: *1. The Annals of Applied Probability, Vol. 16, 423-447, 2006 (with A. Rosalsky), 2. Probability Theory and Related Fields, Vol. 148, 5-35, 2010 (with W. Liu and A. Rosalsky), and 3. Journal of Multivariate Analysis, Vol. 111, 256-270, 2012 (with Y. Qi and A. Rosalsky). The successful completion of my proposed work would be an important step in increasing our understanding of the asymptotic behavior of the largest entry of a sample correlation matrix in very general and applicable situations.**The results related to this proposal will be novel and significant insofar as they will extend, generalize, and refine earlier work in the literature. All results will be formalized in papers for publication in major academic journals.

我的研究建议致力于一般限制概率和渐近统计的定理，并在其应用于各种各样的问题上。**我们生活在一个随机的世界中。概率理论是与随机现象分析有关的数学分支。极限理论是概率和统计的核心，在科学或社会科学的几乎每个分支中都扮演着中心阶段独立随机变量的迭代对数已赋予现代概率理论。它们已在许多方向上扩展和概括为更一般的随机过程和随机措施，它们已成为渐近统计的基础。渐近统计或大型样本理论是评估估计器和统计检验性质的通用框架。在此框架内，通常假定样本量n无限期增长，并且由于样本量n倾向于无穷大。 - 关于几乎确定和随机过程的趋势的研究兴趣，尤其是在迭代对数的定律中，大量定律，中心限制定理，较大和中等偏差的概率以及经典定理中的精确渐近差异的概率以及精确的渐近性定理。 - 值或Banach空间值的随机过程。目的是研究经典限制结果的改进，并开发一些新的方法来证明几乎确定的随机过程的融合，并继续我以前的研究工作，即在概率上使用现代的随机过程技术并开发一些新的过程随机过程的概率不平等，以找到条件，在这些条件下，几乎确定的和弱的收敛性对于随机过程存在并研究这种收敛的统计应用。 L统计数据，U统计量，重采样方法和高维数据分析问题，例如样本相关矩阵的最大进入，有条件密度和模式使用截断和审查的数据估计等等。例如，由统计假设激励测试问题，样品相关矩阵最大进入的渐近行为已在近年来进行了广泛的研究，其中包括我的三篇裁量期刊文章： *1。应用概率的年鉴，第1卷。 16，423-447，2006（与A. Rosalsky一起），2。概率理论和相关领域，第1卷。 148，5-35，2010（与W. Liu和A. Rosalsky一起）和3。多变量分析杂志，第1卷。 111，201，2012（与Y. Qi和A. Rosalsky一起）。在非常普遍且适用的情况下，我提出的工作成功完成将是增加对样本相关矩阵最大进入的渐近行为的重要一步。随着它们将扩展，概括和完善文献中的早期工作。所有结果将在主要学术期刊的论文中正式化。