Analysis and Geometry of Markov Chains Diffusion Processes

这个项目致力于两个主题的研究：(I)概率统计中的极限定理，和(Ii)高斯过程的下尾概率和小球概率。极限定理在概率统计的发展中起着重要的作用。主要研究者继续朝这个方向研究，特别是集中在自归一化极限定理上。本文旨在系统地研究独立随机变量自归一化和、Hotling t统计量和学生化U统计量的中等偏差。目的是建立有限三阶矩条件下的Cramer型中偏差定理。由于自归一化中偏差需要很少的矩条件，它们不仅推广了经典的极限定理，而且还提供了更广泛的适用于其他领域，特别是统计学。这项研究还应该帮助我们更好地理解大类统计泛函的行为，因为t-统计量和U-统计量是它们的基石。另一个证明极限定理有用的领域是研究随机代数和三角多项式的实零点。这种具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概率和小球概率。在估计在天气预报、自然灾害预报和经济指数等具有基本重要性的地区发生罕见事件的可能性时，通常会出现这种类型的概率。目标之一是开发新的方法来估计小球和较低的尾巴概率。重点研究了平稳高斯过程中高维布朗单的小球概率和较低的尾概率。研究人员还打算研究一类新引入的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转性质，但是无穷可微的。人们相信，这一新的高斯过程家族将被证明在许多其他领域作为数学模型是有用的。这个项目致力于两个主题的研究：(I)概率统计中的极限定理，和(Ii)高斯过程的下尾概率和小球概率。极限定理在概率统计的发展中起着重要的作用。希望这项研究的第一部分可以导致概率统计中的自归一化极限理论的发展，而研究的第二部分可以提供关于高斯随机过程以及关于我们的随机环境的重要的新知识。

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.