Probability Measures on Vector Spaces: Theory and Applications

概率在现代科学中的应用经常涉及到对具有许多分量（维数）的随机量的研究，甚至可能是几何性质的随机量。因此，他们需要概率估计和极限定理，适用于随机集，或者是无维的（因此，在本质上，无限维）。在调查员以前的工作中的一个主要主题，在目前的研究中，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价于近似理论中的问题。这种联系导致解决了一个长期存在的问题，在近似理论，部分拟议的研究涉及重要的未解决的类似问题。另一个例子是研究统计力学的吉布斯条件原理，用于无穷多个分量的统计。要开始处理这种类型的问题之一，需要非对数估计的大偏差概率是无量纲的。这些估计严重依赖于支配点和一个合适的概率表示公式。要考虑各种条件极限定理。表现出这些一般功能的其他问题也提出了，并与经典的几何，分析和统计。这项工作包括进一步的非对数大偏差概率的独立随机向量的部分和，调查的支配点在一个更一般的设置，以及应用这些结果的条件极限定理的无限维统计。随机过程的随机样本的极限集，以及相关的覆盖问题将被检查，和一个主要的重点将是进一步研究小球概率和非经典函数的迭代对数适用于占领措施之间的联系。关于向量值部分和，集群集，小球概率，自归一化部分和，布朗运动路径的凸壳极限定理的问题也被考虑。

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.