Lower Tail Probabilities and Limit Theorems in Probability and Statistics

该项目致力于研究两个主题：（i）概率和统计中的极限定理，以及（ii）高斯过程的下尾和小球概率。极限定理在概率论和统计学的发展中起着基础性的作用。主要研究人员继续他的研究在这个方向上一般，特别是专注于自我规范化极限定理。研究者打算系统地研究独立随机变量的自归一化和、Hotelling t统计量和学生化U $统计量的中等偏差。目的是在有限三阶矩条件下建立一个Cramer型中偏差定理。由于自正规化中偏差只需要很少的矩条件，因此它不仅推广了经典的极限定理，而且在其它领域，特别是统计学中也有更广泛的应用.该研究还有助于我们更好地理解大类统计泛函的行为，因为t-统计量和U-统计量是它们的构建块。极限定理证明有用的另一个领域是随机代数和三角多项式的真实的零点的研究。这种随机系数的多项式出现在许多学科中，它们的行为是统计学家，工程师，经济学家和数学家感兴趣的。第二个主题的主要重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，这些事件在天气预测，自然灾害预测和经济指数等领域具有根本重要性。目标之一是开发新的方法来估计小球和下尾概率。重点是具体的小球概率的布朗单在高维和较低的尾部概率平稳高斯过程。研究人员还打算研究一个新引入的家庭的高斯过程，具有相同的缩放和时间反演性质的布朗运动，但无限可微的基本样本属性。据信，这一新的高斯过程族将被证明在许多其他领域作为数学模型是有用的。 该项目致力于研究两个主题：（i）概率和统计中的极限定理，以及（ii）高斯过程的下尾和小球概率。极限定理在概率论和统计学的发展中起着基础性的作用。希望这项研究的第一部分可能会导致概率和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有关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.