Mathematical Sciences: Limit Theorems for Function-Indexed Empirical Processes

数学科学：函数索引经验过程的极限定理

• 批准号: 8702906
• 负责人: Kenneth Alexander
• 金额: $ 3.82万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-15 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702906&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Limit; Theorems; Function

The first problem this research will consider is that of limit theorems for empirical processes. Considerable progress has been made in recent years towards solving the problem of necessary and sufficient conditions on probability measures and classes of functions for which the corresponding empirical process satisfies the central limit theorem. Some of these advances were made by the investigator himself. He will continue to look for the final solution of the problem. The second problem concerns the law of iterated logarithms for Banach space valued random variables. A set of three conditions is found to be both necessary and sufficient but one of these conditions amounts to proving a weaker version of the original problem. The investigator will attack this reduced problem. The third and smaller part of the research will deal with percolation theory. Among the topics to be considered will be continuity of the percolation probability, bounds on critical exponents and finiteness of the expected size of cluster containing zero.

本研究将考虑的第一个问题是经验过程的极限定理问题。 近年来，在解决相应的经验过程满足中心极限定理的概率测度和函数类的充要条件问题方面已经取得了相当大的进展。 其中一些进展是由研究者本人取得的。 他将继续寻找问题的最终解决方案。 第二个问题涉及巴拿赫空间随机变量的迭代对数定律。 发现一组三个条件既必要又充分，但其中一个条件相当于证明原始问题的较弱版本。 研究者将解决这个简化的问题。 研究的第三部分（也是较小的一部分）将涉及渗透理论。 要考虑的主题包括渗滤概率的连续性、临界指数的界限以及包含零的簇的预期大小的有限性。