Probability Measures of Vector Spaces; Basic Results and Applications

[703740] Kuelbs首席研究员将继续他在概率论方面的研究。它包含了无限维情况下独立随机向量通过支配点的部分和的非对数大偏差概率，并将这些结果应用于某些无限维统计量的Gibbs条件作用原理。过程随机样本的极限集，以及相关的覆盖问题将被检查，一个主要的焦点将是进一步检查小球概率和近似理论中的度量熵问题之间的联系。对于高斯测度，这种联系在许多问题领域是有用的，但从几个角度提出了更详细的检查。有关向量值部分和，经验过程，自归一化部分和，以及布朗运动凸壳极限定理的问题也将被考虑。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）。在研究者以前的工作和目前提出的研究中，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。要开始处理这类问题，就需要无维的大偏差概率的非对数估计。这些估计是最近才得到的，目前正在将它们应用于吉布斯条件反射。还需要考虑显示这些一般特征的其他问题，并将其与经典几何、分析和统计联系起来。

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.