Mathematical Sciences: Probability Measures on Vector Spaces; Basic Results and Application

数学科学：向量空间上的概率测度；

• 批准号: 9400024
• 负责人: James Kuelbs
• 金额: $ 8万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400024&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Probability; Measures; Vector

This continues research involving limit theorems for real and vector valued random variables. Problems related to the central limit theorem and the law of the iterated logarithm motivated by trimming and self-normalizations will be studied. Limit sets for random samples of processes, as well as coverage problems will be examined. A primary focus will be to further examine the link between the small ball problem for Gaussian measures and metric entropy problems. Empirical processes and precise large deviation probabilities will also be considered. A central theme in probability, and throughout its diverse applications to problems in engineering, physics, and statistics, is the study of limit theorems for partial sums of independent random variables in both finite and infinite dimensional settings. This research place emphasis on the infinite dimensional situation and involves a blend of functional analysis and probability theory. Current work has fallen into several different areas, and includes small ball and large deviation probabilities for Gaussian measures, limit sets for random samples and related coverage problems, and Gaussian chaos applications to multiple Wiener-Ito stochastic integrals.

这将继续研究实值和向量值随机变量的极限定理。将研究与中心极限定理和重对数律有关的问题，这些问题是由裁剪和自归一化引起的。将研究随机过程样本的极限集，以及覆盖问题。主要的焦点将是进一步研究高斯测量的小球问题和度量熵问题之间的联系。经验过程和精确的大偏差概率也将被考虑。概率论的一个中心主题，以及它在工程、物理和统计学问题上的各种应用，都是研究独立随机变量在有限维和无限维环境下的部分和的极限定理。本研究侧重于无限维情形，涉及到泛函分析和概率论的融合。目前的工作分为几个不同的领域，包括高斯测量的小球概率和大偏差概率，随机样本的极限集和相关的覆盖问题，以及高斯混沌在多重Wiener-Ito随机积分中的应用。