Gaussian Methods and Probability Estimates of Rare Events

本研究以发展高斯方法研究罕见事件为中心，主要依赖于一个正随机量的小值。主要目标是扩展对相关主题的理解，并在系统研究各种技术和应用的基础上建立一个一般理论。最近完成的小球概率和度量熵问题之间的联系允许应用泛函分析的工具和结果，为估计高斯过程的下界提供了目前最强大的方法。反过来，它提出了许多与概率论和几何泛函分析应用有关的进一步问题。类似的关系将研究其他随机过程，如稳定过程和高斯混沌。关于依赖小值的罕见事件的概述统一了数学中不同领域的许多问题。高斯方法在降低尾部概率、布朗追踪模型和首次退出时间方面的非常成功的应用将扩展到对随机多项式、平衡向量和Hadamard矩阵的零的详细研究。本研究的主要焦点是更好地理解与高斯过程和其他在许多应用中作为模型的罕见随机现象。这类问题往往是在估计罕见事件发生的几率时产生的，而这些罕见事件发生在天气、经济指数、流行病等具有根本重要性的地区。本研究将提高我们对罕见随机事件的认识，并为研究我们的随机环境提供基本工具。

0204513Li This research centers on developing Gaussian methods for the study of rare events, mainly depending on small values of a positive random quantity. The major objectives are to extend the understanding of related topics and build a general theory based on systematic study of various techniques and applications. The recent completion of the connection between small ball probabilities and metric entropy problems allows applications of tools and results from functional analysis to provide the most powerful method available at this time for estimating the lower bound for Gaussian processes. In turn, it suggests many further questions connected to applications in probability theory and geometric functional analysis. Similar relationships will be studied for other random processes such as stable processes and Gaussian chaos. The overview on rare events depending on small values unifies many problems from diverse areas in mathematics. The very successful applications of Gaussian methods to lower tail probabilities, the Brownian pursuit models and the first exit times will be expanded to a detailed study of zeros of random polynomials, balancing vectors, and Hadamard matrices. The primary focus of this research is a better understanding of rare random phenomena related to Gaussian processes and others 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 should improve our understanding of rare random events and provide basic tools for the study of our random environment.