Probability Measures of Vector Spaces; Basic Results and Applications

9703740 Kuelbs首席研究员将继续他在概率论的研究。 它包括非对数的大偏差概率的独立随机向量的部分和通过在无限维设置的主导点，以及这些结果的应用吉布斯条件的原则，某些无限维统计。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近似理论中的小球概率和度量熵问题之间的联系。 对于高斯措施，这种联系在一些问题领域是有用的，但从几个角度提出了更详细的检查。关于向量值部分和，经验过程，自正规化部分和，布朗运动的凸壳极限定理的问题也将被考虑。 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）。一个主要的主题，在调查员以前的工作，并在目前提出的研究解决这两个问题，在各种设置。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。为了开始处理这类问题，需要无量纲的大偏差概率的非对数估计。 这种估计是最近获得的，他们的应用吉布斯空调正在启动。表现出这些一般特征的其他问题也要考虑，并与经典几何，分析和统计学。

9703740 Kuelbs The principal investigator will continue his research in probability theory. It includes non-logarithmic large deviation probabilities for partial sums of independent random vectors via dominating points in the infinite dimensional setting, and also the application of these results to a Gibbs conditioning principle for certain infinite dimensional statistics. Limit sets for random samples of 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 metric entropy problems in approximation theory. For Gaussian measures, this linkage has been useful in a number of problem areas, but a more detailed examination from several points of view is proposed. Problems concerning vector valued partial sums, empirical processes, self-normalized partial sums, and limit theorems for convex hulls for Brownian motion are also to be considered. 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 currently proposed 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 estimat es of large deviation probabilities which are dimension free. Such estimates were obtained recently, and their application to Gibbs conditioning is being initiated. Additional problems exhibiting these general features are also to be considered, and connect with classical geometry, analysis, and statistics.