Self-Normalized Limit Theorems and Small Ball Probabilities

9802451邵本项目主要研究概率统计中的两个主题，即自归一化极限定理和小球概率。 第一个主题是致力于研究一般自正规化过程的极限定理，特别是自正规化部分和。 经典极限定理中的正规化常数通常是真实的数列。 矩条件或其他相关假设是许多经典极限定理的充分必要条件。 然而，当正规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，不需要矩条件的大偏差类型的结果。 自正规化的重对数律对正态或稳定律的吸引域中的所有分布仍然有效。 这表明自规范化比确定性规范化保留了更好的性质。 这也提出了许多进一步的问题，如什么是自规范化近似的收敛速度，什么是自规范化的迭代对数律的必要和充分条件，寻找自规范化的截尾和截尾和的尾概率，以及寻找独立但不一定同分布的随机变量的自规范化极限定理。 第二个主题是关于高斯过程的小球概率，它在许多应用中用作模型。 小球概率为罕见事件提供了精确的估计。 这部分研究的一个主要重点是更好地理解与高斯过程相关的罕见随机现象，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。 这部分的研究涉及到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-统计量是它们的基石。 小球问题经常出现在估计罕见事件发生的概率时，这些事件在天气、经济指数和流行病等具有根本重要性的领域中发生。 这项研究的第一部分可能会导致一个新的极限理论的概率和统计的发展，而第二部分的研究可能会提供重要的新知识高斯过程以及我们的随机环境。

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.