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.

摘要研究者计划研究三个不同领域的问题。这些是(1)独立随机变量和的行为的普遍结果，(2)局部经验过程和(3)经验增量的分形性质。区域(1)涉及部分和及其增量在很少或没有假设的情况下的迭代对数规律的研究。区域(2)涉及一种称为局部经验过程的新数学实体的研究，它是控制密度估计器、回归函数估计器和尾指数估计器的随机机制。区域(3)揭示了各种随机过程的增量的一些意想不到的分形性质。本文研究的是一类随机过程的极限行为及其应用。一个随机过程在经过长时间观察后，往往会表现出稳定的性质。经典的例子是随机抛硬币的过程。如果硬币是均匀的，那么长时间内正面的平均次数肯定会任意接近1/2。这个事实是大数定律的一个特例。统计学家利用这一规律，以大概率接近感兴趣的总体参数的数据样本为基础构建估计器。在这个项目中将要调查的领域应该对建造这样的估算器有一些有用的应用。