General Limit Theorems in Probability with Applications to Statistics

我的研究建议致力于概率统计和渐近统计中的一般极限定理，以及它们在各种问题中的应用。 我们生活在一个随机的世界里。概率论是研究随机现象分析的数学分支。极限理论是概率和统计学的核心，在几乎所有科学或社会科学的分支中都发挥着中心作用，包括天气预报、地理学等。大数定律、中心极限定理和独立随机变量的重对数律等结果形成了现代概率理论。它们已经在许多方向上被推广和推广，其中包括更一般的随机过程和随机测度，它们已经成为渐近统计的基础。渐近统计或大样本理论是评估估计量和统计检验性质的通用框架。在这个框架内，通常假设样本大小n无限增长，并且随着样本大小n趋于无穷大，统计过程的性质在极限内被评估。 这项研究建议的第一个重点涉及我对随机过程的几乎必然和弱收敛的长期研究兴趣，特别是在实值或Banach空间值随机过程的经典极限定理中的重对数律、大数定律、中心极限定理、大偏差和中偏差的概率以及精确渐近性。目的是研究经典极限结果的精化，发展一些新的方法来证明随机过程的几乎必然和弱收敛，并继续我以前的研究工作，即在概率论中使用现代随机过程技巧，发展一些新的随机过程的概率不等式，以便找到随机过程几乎必然和弱收敛的条件，并研究这种收敛的统计应用。 第二个重点将集中在研究与分层模型、L统计、U统计、重采样方法和高维数据分析问题有关的统计应用中的渐近行为，例如样本相关矩阵的最大项、截断和删失数据的条件密度和模式的估计等。例如，出于统计假设检验问题的动机，近年来广泛地研究了样本相关矩阵的最大项的渐近行为，包括我的三篇参考期刊文章： 1.《应用概率年鉴》，第16卷，423-447卷，2006年(与A.Rosalsky合著)；2.《概率论与相关领域》，第148卷，5-35卷，2010年(与刘伟、A.Rosalsky合著)；3.《多元分析杂志》，第111卷，256-270卷，2012年(与齐亚英、A.Rosalsky合著)。我提议的工作的成功完成将是增加我们对样本相关矩阵最大项在非常一般和适用的情况下的渐近行为的理解的重要一步。 与这一提议相关的结果将是新颖和有意义的，因为它们将扩展、概括和完善文献中的早期工作。所有结果都将以论文的形式正式发表在主要学术期刊上。

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.