Analysis and Geometry of Markov Chains Diffusion Processes

本项目主要研究两个课题：(1)概率论和统计学中的极限定理，(2)高斯过程的下尾概率和小球概率。极限定理在概率论和统计学的发展中起着重要的作用。首席研究员在这个方向上继续他的研究，特别关注自归一化极限定理。研究者打算系统地研究独立随机变量的自归一化和、霍特林t统计量和学生化u统计量的适度偏差。目的是建立有限三阶矩条件下的Cramer型中等偏差定理。由于自归一化的中等偏差需要很少的矩条件，它们不仅扩展了经典极限定理，而且在其他领域，特别是统计领域提供了更广泛的适用性。这项研究还应该帮助我们更好地理解大量统计函数的行为，因为t统计量和u统计量是它们的组成部分。极限定理被证明有用的另一个领域是对随机代数多项式和三角多项式的实零的研究。这种带有随机系数的多项式出现在许多学科中，它们的行为引起了统计学家、工程师、经济学家和数学家的兴趣。第二个主题的主要焦点是估计高斯过程的下尾和小球概率。这些类型的概率通常出现在估计罕见事件发生的可能性时，这些事件在诸如天气预报、自然灾害预测和经济指数等方面具有根本重要性。目标之一是开发估计小球和低尾概率的新方法。重点是在高维布朗表的小球概率和平稳高斯过程的低尾概率。研究者还打算研究新引入的一类高斯过程的基本样本性质，这些过程具有与布朗运动相同的标度和时间反演性质，但具有无限可微性。人们相信，这个新的高斯过程族将在许多其他领域被证明是有用的数学模型。本项目主要研究两个课题：(1)概率论和统计学中的极限定理，(2)高斯过程的下尾概率和小球概率。极限定理在概率论和统计学的发展中起着重要的作用。希望本研究的第一部分可能会导致概率论和统计学中自归一化极限理论的发展，而第二部分的研究可以提供关于高斯随机过程以及我们的随机环境的重要新知识。

This project is devoted to the study of two topics: (i) limit theorems in probability and statistics, and (ii) lower tail and small ball probabilities of Gaussian processes. Limit theorems play a fundamental role in the development of probability and statistics. The principal investigator continues his study in this direction in general, focusing on self-normalized limit theorems in particular. The investigator intends to systematically study moderate deviations for self-normalized sums of independent random variables, for Hotelling's t-statistic and for studentized U-statistic. The objective is to establish a Cramer type moderate deviation theorem under a finite third moment condition. Since the self-normalized moderate deviations require few moment conditions, they not only extend classical limit theorems but also provide much wider applicability to other fields, particularly to statistics. The study should also help us better understand the behavior of large classes of statistical functionals since the t-statistic and U-statistic are their building blocks. Another area where limit theorems prove useful is the study of the real zeros of random algebraic and trigonometric polynomials. Such polynomials with random coefficients arise in many disciplines and their behavior is of interest to statisticians, engineers, economists, and mathematicians. The primary focus of the second topic is on estimating lower tail and small ball probabilities for Gaussian processes. These types of probabilities often arise in estimating the chances of rare events occurring in areas where such events are of fundamental importance such as weather prediction, natural disaster prediction and economic indices. One of the objectives is to develop new methods of estimating small ball and lower tail probabilities. The focus is specifically on small ball probabilities of the Brownian sheet in high dimensions and lower tail probabilities for stationary Gaussian processes. The investigator also intends to study basic sample properties for a newly introduced family of Gaussian processes which have the same scaling and time inversion properties as the Brownian motion but are infinitely differentiable. It is believed that this new family of Gaussian processes would prove useful in many other fields as mathematical models. This project is devoted to the study of two topics: (i) limit theorems in probability and statistics, and (ii) lower tail and small ball probabilities of Gaussian processes. Limit theorems play a fundamental role in the development of probability and statistics. It is hoped that the first part of this research may lead to the development of a self-normalized limit theory in probability and statistics, while the second part of the research could provide significant new knowledge about Gaussian random processes as well as about our random environments.