Mathematical Sciences: Gaussian Measures and Small Ball Probabilities

摘要本研究涉及三个密切相关的概率论领域，即高斯等周不等式、小球概率和高斯过程的极限定理。研究了限制集上的高斯等周不等式。这导致了精确的估计，并扩展了等周工具在各种情况下的适用范围。小球概率为罕见事件提供了精确的估计，到目前为止，只有几种通用的方法可用。最近发现的小球概率和度量熵问题之间的联系，为这一领域提供了一些启示，并提出了许多进一步的问题，特别是在近似理论中的应用。许多概率估计是由强极限定理的研究引起的。极限定理的研究主要围绕高斯过程的极限行为展开。本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，这些现象在许多应用中作为模型。这类问题往往是在估计罕见事件发生的几率时产生的，而这些事件在天气、经济指数、流行病等具有根本重要性的领域发生。这项研究可以在概率论方面带来重要的新知识，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.