Mathematical Sciences: Gaussian Measures and Small Ball Probabilities

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9503458 Li Abstract This research involves work in three closely related areas of probability, namely, Gaussian isoperimetric inequalities, small ball probabilities, and limit theorems for Gaussian processes. Gaussian isoperimetric inequalities are studied for restricted sets. This leads to sharp estimates and extends the domain of applicability of the isoperimetric tool in a variety of settings. Small ball probabilities provide sharp estimates for rare events and so far there are only a few general methods available. The recent discovery of the connection between small ball probabilities and metric entropy problems sheds some light in this area and suggests many fupther questions, in particular, the applications to approximation theory. Many of the probability estimates are motivated by the study of strong limit theorems. The research in limit theorems is centered around the limiting behaviors of Gaussian processes. The primary focus of this research is a better understanding of rare random phenomena related to Gaussian processes which serve as models in many applications. These types of 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, epidemics etc. This research can lead to significant new knowledge in probability and provide basic tools for the study of our random environment.