Self-Normalized Limit Theorems and Small Ball Probabilities

这个项目专注于概率和统计学中的两个主题，即自归一化极限定理和小球概率。第一个主题致力于研究一般自正规过程的极限定理，特别是自正规部分和的极限定理。经典极限定理中的归一化常数通常是实数序列。矩条件或其他相关假设是许多经典极限定理的充要条件。然而，当归一化常数是随机变量序列时，情况变得非常不同。最近发现的自归一化大偏差表明，大偏差类型的结果不需要矩条件。一个自归一化的重对数律对正态或稳定律吸引域中的所有分布仍然有效。这表明，自归一化比确定性归一化保留了更好的性质。这也提出了许多进一步的问题，例如，自归一化逼近的收敛速度是多少，重对数律的自归一化的充要条件是什么，自归一化截尾和的尾概率的寻找，以及独立但不一定同分布的随机变量的自归一化极限定理的寻找。第二个主题涉及在许多应用中用作模型的高斯过程的小球概率。小球的概率为罕见事件提供了准确的估计。这部分研究的一个主要焦点是更好地理解与高斯过程有关的罕见随机现象。自归一化和与著名的学生t统计量和学生化的U统计量密切相关。这部分研究涉及确定何时可以安全地使用t统计量和U统计量。这项研究将有助于理解大类统计泛函的行为，因为t-和U-统计是它们的基石。在估计罕见事件在天气、经济指数和流行病等具有根本重要性的地区发生的可能性时，经常会出现小球问题。这项研究的第一部分可能会导致概率统计中一种新的极限理论的发展，而第二部分的研究可能会提供关于高斯过程和我们的随机环境的重要的新知识。

9802451ShaoThis project focuses on two topics in probability and statistics, namely, self-normalized limit theorems and small ball probabilities. The first topic is devoted to the study of limit theorems for self-normalized processes in general, and for self-normalized partial sums in particular. The normalizing constants in classical limit theorems are usually sequences of real numbers. Moment conditions or other related assumptions are necessary and sufficient for many classical limit theorems. However, the situation becomes very different when the normalizing constants are sequences of random variables. The recent discovery of the self-normalized large deviations shows that no moment conditions are needed for a large deviation type result. A self-normalized law of the iterated logarithm remains valid for all distributions in the domain of attraction of a normal or stable law. This reveals that the self-normalization preserves much better properties than deterministic normalization does. This also suggests many further questions, such as what is the rate of convergence of self-normalized approximation, what are necessary and sufficient conditions for the self-normalized law of the iterated logarithm, finding tail probabilities of self-normalized trimmed and censored sums, and finding self-normalized limit theorems for independent but not necessarily identically distributed random variables. The second topic concerns small ball probabilities for Gaussian processes which serve as models in many applications. Small ball probabilities provide sharp estimates for rare events. A primary focus of this part of research is a better understanding of rare random phenomena related to Gaussian processes.The self-normalized sums are closely related to the celebrated ``Student t-statistic" and studentized ``U-statistic". This part of the research is related to determining when the t-statistic and U-statistic can safely be used. This study will help to understand the behavior of large classes of statistical functionals since t- and U-statistics are their building blocks. The small ball 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, and epidemics. The first part of this research may lead to the development of a new limit theory in probability and statistics while the second part of the research may provide significant new knowledge about Gaussian processes as well as about our random environments.