Probability Measures on Vector Spaces: Theory and Applications

向量空间的概率测度：理论与应用

• 批准号: 0071700
• 负责人: James Kuelbs
• 金额: $ 9万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071700&HistoricalAwards=false
• 关键词: Probability; Measures; Vector; Spaces; Theory

Applications of probability in modern science frequently involve the study of random quantities with many components (dimensions), or perhaps even of a geometric nature. Thus they require probability estimates and limit theorems which are applicable to random sets, or which are dimension free (hence, in essence, infinite dimensional). A major theme in the investigator's previous work, and in much of the current research, addresses both of these issues in a variety of settings. As a first example consider the link between small ball probabilities and metric entropy problems, which showed certain probability estimates are equivalent to problems in approximation theory. This link led to the solution of a long standing problem in approximation theory, and portions of the proposed research involve important unsolved analogues of this problem. Another example is the study of the Gibbs conditioning principle of statistical mechanics for statistics with infinitely many components. To begin to handle this type of problem one needs non-logarithmic estimates of large deviation probabilities which are dimension free. These estimates depend critically on dominating points and a suitable representation formula for the probabilities. A variety of conditional limit theorems are to be considered. Additional problems exhibiting these general features are also proposed, and connect with classical geometry, analysis, and statistics. This work includes further non-logarithmic large deviation probabilities for partial sums of independent random vectors, an investigation of dominating points in a more general setting, and the application of these results to conditional limit theorems for infinite dimensional statistics. Limit sets for random samples of stochastic processes, as well as related coverage problems will be examined, and a primary focus will be to further examine the link between small ball probabilities and non-classical functional laws of the iterated logarithm applicable to occupation measures. Problems concerning vector valued partial sums, cluster sets, small ball probabilities, self-normalized partial sums, and limit theorems for convex hulls of Brownian motion paths are also to be considered.

概率论在现代科学中的应用经常涉及对具有许多组成部分（维度）的随机数量的研究，甚至可能涉及几何性质。因此，它们需要概率估计和极限定理，这些定理适用于随机集合，或者是无维的（因此，本质上是无限维的）。在研究者之前的工作和目前的研究中，一个主要的主题是在各种环境中解决这两个问题。作为第一个例子，考虑小球概率和度量熵问题之间的联系，这表明某些概率估计等同于近似理论中的问题。这一联系导致了近似理论中一个长期存在的问题的解决，而部分提出的研究涉及到这个问题的重要的未解决的类似问题。另一个例子是研究具有无限多分量的统计力学的吉布斯条件反射原理。为了开始处理这类问题，需要对无维的大偏差概率进行非对数估计。这些估计主要取决于主导点和合适的概率表示公式。各种条件极限定理将被考虑。还提出了具有这些一般特征的附加问题，并与经典几何、分析和统计相联系。这项工作包括独立随机向量部分和的非对数大偏差概率，更一般设置下支配点的研究，以及将这些结果应用于无限维统计的条件极限定理。将审查随机过程的随机样本的极限集以及相关的覆盖问题，主要重点将是进一步审查小球概率与适用于职业措施的迭代对数的非经典泛函律之间的联系。有关向量值部分和、聚类集、小球概率、自归一化部分和和布朗运动路径凸壳极限定理的问题也将被考虑。