Lower Tail Probabilities and Limit Theorems in Probability and Statistics

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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.