Small Ball Probabilities and Their Applications

本研究的中心是估计小球型罕见事件的概率。两个主要目标是开发新的方法估计小球概率的高斯和密切相关的随机过程，并系统地研究现有的技术和应用，这是分散在各个主题。最近完成的小球概率和度量熵问题之间的连接，允许应用工具和功能分析的结果，以提供一个最强大的方法在下界估计高斯过程。反过来，它提出了许多与概率论和近似论中的应用有关的进一步问题。PI打算将这项研究扩展到其他随机过程，如高斯混沌和稳定过程。这项研究应该导致显着的小球概率的新知识，并为随机过程的研究提供了基本的工具。这项研究的主要重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。这些类型的问题往往出现在估计的机会，罕见的事件发生在这些事件是根本的重要性，如天气，经济指数，流行病等领域，这项研究应提高我们的理解罕见的随机事件，并提供基本的工具，为我们的随机环境的研究。

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.