Mathematical Sciences: General Limit Theorems in Probability and Local Empirical Processes

数学科学：概率中的一般极限定理和局部经验过程

• 批准号: 9503665
• 负责人: Uwe Einmahl
• 金额: $ 4万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-15 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503665&HistoricalAwards=false
• 关键词: Mathematical; Sciences; General; Limit; Theorems

9503665 Einmahl Abstract This proposal addresses questions from three different areas: (1) Universal Results on the Behavior of Sums of Independent Random Variables (2) Local Empirical Processes (3) A General Law of the Iterated Logarithm in Banach space. The work in area (1) is motivated by some recent universal results on the almost sure behavior of sums of independent random variables. Contrary to the classical results in this direction, these results do not require moment type conditions. The main difficulty is to find suitable norming and centering sequences which are specific to the particular situations under investigation. Among other things, quantile transformations will be used. The work from area (2) is mainly devoted to the limit behavior of local empirical processes which have useful applications in density and regression function estimation. The third part of this proposal finally addresses a number of questions in connection with a recent general law of the iterated logarithm in Banach space. This proposal addresses a number of questions on limit theorems in probability and empirical processes. Limit theorems in probability such as the central limit theorem, and laws of large numbers provide the basis for many widely used statistical techniques to handle large data sets. Some of these results, however, can only be applied under relatively restrictive conditions. Over the last decade, powerful methods have been developed in probability. It is proposed to use these new techniques to further extend the applicability of the basic limit theorems, and, at the same time, to gain some insight into situations, where the classical techniques have failed. Some applications to statistical estimation theory will be considered as well, where the main emphasis will be on so-called empirical processes.

摘要本文讨论了三个不同领域的问题：(1)独立随机变量和行为的一般结果；(2)局部经验过程；(3)Banach空间中迭代对数的一般规律。领域(1)的工作是由最近关于独立随机变量和的几乎确定行为的一些普遍结果所激发的。与这个方向的经典结果相反，这些结果不需要矩型条件。主要的困难是找到适合于所研究的特定情况的规范和定心序列。除其他事项外，还将使用分位数转换。区域(2)的工作主要致力于局部经验过程的极限行为，这些过程在密度和回归函数估计中有有用的应用。最后，本提案的第三部分解决了与巴拿赫空间中最近的迭代对数一般定律有关的一些问题。本文提出了概率论和经验过程中关于极限定理的若干问题。概率论中的极限定理，如中心极限定理和大数定律，为处理大型数据集的许多广泛使用的统计技术提供了基础。然而，其中一些结果只能在相对有限的条件下应用。在过去的十年里，概率论发展出了强大的方法。利用这些新技术进一步扩展了基本极限定理的适用性，同时也对经典技术无法解决的问题有了新的认识。也将考虑统计估计理论的一些应用，其中主要的重点将是所谓的经验过程。