Mathematical Sciences: Probability Measures on Vector Spaces; Basic Results and Application

这继续研究涉及极限定理的真实的和向量值随机变量。 中心极限定理和重对数律的相关问题将被研究，这些问题是由修剪和自规范化引起的。 过程的随机样本的极限集，以及覆盖问题将被检查。 一个主要的重点将是进一步研究之间的联系的小球问题的高斯措施和度量熵的问题。 经验过程和精确的大偏差概率也将被考虑。 概率论的一个中心主题，以及它在工程、物理和统计问题中的各种应用，是研究有限维和无限维情况下独立随机变量部分和的极限定理。 本研究着重于无限维情形，并融合了泛函分析和概率论。 目前的工作已经落入几个不同的领域，包括小球和大偏差概率高斯措施，限制设置随机样本和相关的覆盖问题，和高斯混沌应用多个维纳-伊藤随机积分。

This continues research involving limit theorems for real and vector valued random variables. Problems related to the central limit theorem and the law of the iterated logarithm motivated by trimming and self-normalizations will be studied. Limit sets for random samples of processes, as well as coverage problems will be examined. A primary focus will be to further examine the link between the small ball problem for Gaussian measures and metric entropy problems. Empirical processes and precise large deviation probabilities will also be considered. A central theme in probability, and throughout its diverse applications to problems in engineering, physics, and statistics, is the study of limit theorems for partial sums of independent random variables in both finite and infinite dimensional settings. This research place emphasis on the infinite dimensional situation and involves a blend of functional analysis and probability theory. Current work has fallen into several different areas, and includes small ball and large deviation probabilities for Gaussian measures, limit sets for random samples and related coverage problems, and Gaussian chaos applications to multiple Wiener-Ito stochastic integrals.