This 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.
