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: 9504758
• 负责人: David Mason
• 金额: $ 7.5万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504758&HistoricalAwards=false
• 关键词: Mathematical; Sciences; General; Limit; Theorems

9504758 Mason Abstract The investigator plans to study problems in three different areas. These are (1) universal results for the behavior of sums of independent random variables, (2) local empirical processes and (3) the fractal nature of empirical increments. Area (1) concerns the study of the law of the iterated logarithm for partial sums and their increments that hold under little or no assumptions. Area (2) involves the investigation of a new mathematical entity called the local empirical process, which is the random mechanism governing density estimators, regression function estimators and tail index estimators. Area (3) deals with the uncovering of the somewhat unexpected fractal nature of the increments of various random processes. The proposed research is concerned with the study of the limit behavior of certain random processes and its applications. Very often a random process will exhibit stable properties as it is observed over a long period. The classical example is the random process of tossing a coin. If the coin is fair, the long time average number of heads will with certainty get arbitrarily close to 1/2. This fact is a special case of the law of large numbers. Statisticians take advantage of this law to construct estimators based on a large sample of data that are close with high probability to the population parameter of interest. The areas that will be investigated in this project should have a number of useful applications to the construction of such estimators.

9504758梅森摘要调查员计划研究三个不同领域的问题。这些是(1)独立随机变量和行为的普遍结果，(2)局部经验过程和(3)经验增量的分形性。领域(1)涉及对部分和及其增量的重对数律的研究，这些部分和及其增量在很少或不存在假设的情况下成立。区域(2)涉及一种称为局部经验过程的新的数学实体的研究，它是支配密度估计、回归函数估计和尾部指数估计的随机机制。区域(3)涉及揭示各种随机过程的增量的有点出乎意料的分形性。本文主要研究某些随机过程的极限行为及其应用。通常情况下，随机过程会表现出稳定的性质，因为它是在长时间内观察到的。一个经典的例子是掷硬币的随机过程。如果硬币是公平的，长期平均头数肯定会任意接近1/2。这是大数定律的一个特例。统计学家利用这一定律，根据与感兴趣的总体参数有很高概率接近的大样本数据，构建估计器。本项目中将要调查的领域应该在构建这种估计器方面有一些有用的应用。