Probability Measures of Vector Spaces; Basic Results and Applications

9703740首席研究员库尔布斯将继续他在概率论方面的研究。它包括独立随机向量的部分和在无穷维集上通过控制点的非对数大偏差概率，以及这些结果在某些无限维统计量的Gibbs条件原理中的应用。本课程将研究随机过程样本的极限集以及相关的覆盖问题，主要焦点将是进一步研究小球概率和近似理论中的度量熵问题之间的联系。对于高斯测量，这种联系在许多问题领域都很有用，但从几个角度提出了更详细的检查。还将考虑向量值部分和、经验过程、自归一化部分和以及布朗运动凸壳的极限定理等问题。概率在现代科学中的应用经常涉及研究具有许多分量(维度)的随机量，或者甚至可能具有几何性质。因此，它们需要适用于随机集的概率估计和极限定理，或者是无量纲的(因此，本质上是无限维的)。调查员以前的工作和目前提出的许多研究中的一个主要主题在各种情况下都涉及这两个问题。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。这种联系导致了近似理论中一个长期存在的问题的解决，拟议的研究的一部分涉及这个问题的重要的未解决的类似物。另一个例子是研究具有无穷多个分量的统计力学的吉布斯条件原理。要开始处理这类问题，需要无因次的大偏差概率的非对数估计。这样的估计是最近获得的，它们在Gibbs条件作用中的应用正在启动。表现这些一般特征的其他问题也要考虑，并与经典几何学、分析和统计学相联系。

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.