Small Ball Probabilities and Their Applications

这项研究的中心是估计小球类型罕见事件的概率。两个主要目标是发展估计高斯和密切相关随机过程的小球概率的新方法，并系统地研究分布在不同主题的现有技术和应用。最近完成了小球概率和度量熵问题之间的联系，使得泛函分析的工具和结果的应用可以为高斯过程的下界估计提供最强大的方法之一。反过来，它提出了许多与概率论和近似论中的应用有关的更深层次的问题。PI打算将这一研究扩展到其他随机过程，如高斯混沌和稳定过程。这项研究应该会带来关于小球概率的重要新知识，并为随机过程的研究提供基本工具。这项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用作模型的罕见随机现象有关的现象。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几率时，这类问题经常会出现。这项研究应该会提高我们对罕见随机事件的理解，并为研究我们的随机环境提供基本工具。

This research centers on estimating the probability of rare events of small ball type. The two major objectives are to develop new methods of estimating small ball probabilities for Gaussian and closely related random processes, and to systematically study the existing techniques and applications which are spread over various topics. 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 one of the most powerful methods in the lower bound estimate for Gaussian processes. In turn, it suggests many further questions connected to applications in probability theory and approximation theory. The PI intends to extend this study to other random processes such as Gaussian chaos and stable processes. This research should lead to significant new knowledge about small ball probabilities and provide basic tools for the study of random processes. 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.