Mathematical Sciences: Gaussian Measures and Small Ball Probabilities

9503458 LI摘要这项研究涉及三个密切相关的概率领域，即高斯等周不等式、小球概率和高斯过程的极限定理。研究了约束集上的高斯等周不等式。这导致了准确的估计，并扩大了等周工具在各种环境中的适用范围。小球概率为罕见事件提供了准确的估计，到目前为止，只有少数几种通用方法可用。最近发现的小球概率和度量熵问题之间的联系揭示了这一领域的一些曙光，并提出了许多进一步的问题，特别是在近似理论中的应用。许多概率估计都是基于对强极限定理的研究。极限定理的研究主要围绕着高斯过程的极限行为展开。这项研究的主要焦点是更好地理解与高斯过程相关的罕见随机现象，这些过程在许多应用中用作模型。这些类型的问题经常出现在估计罕见事件在天气、经济指数、流行病等具有基本重要性的地区发生的概率时。这项研究可以带来关于概率的重要新知识，并为研究我们的随机环境提供基本工具。

9503458 Li Abstract This research involves work in three closely related areas of probability, namely, Gaussian isoperimetric inequalities, small ball probabilities, and limit theorems for Gaussian processes. Gaussian isoperimetric inequalities are studied for restricted sets. This leads to sharp estimates and extends the domain of applicability of the isoperimetric tool in a variety of settings. Small ball probabilities provide sharp estimates for rare events and so far there are only a few general methods available. The recent discovery of the connection between small ball probabilities and metric entropy problems sheds some light in this area and suggests many fupther questions, in particular, the applications to approximation theory. Many of the probability estimates are motivated by the study of strong limit theorems. The research in limit theorems is centered around the limiting behaviors of Gaussian processes. The primary focus of this research is a better understanding of rare random phenomena related to Gaussian processes which serve as models in many applications. These types of problems often arise in estimating the chances for rare events to occur in areas where such events are of fundamental importance, such as weather, economic indices, epidemics etc. This research can lead to significant new knowledge in probability and provide basic tools for the study of our random environment.