The main objectives of the project are as follows: * Prove the longstanding conjecture on the best constant factor in the Rademacher-Gaussian tail comparison. * Prove another longstanding conjecture, on the asymptotic domination of the Rademacher tail by the Gaussian one. * Consider also the ``asymmetric'' case. * Extend to the case of moderate deviations the result due to Shao et al. on the saddle-point approximation to large-deviation probabilities of a self-normalized sum of independent random variables. * Obtain limit theorems, including Berry-Esseen-type bounds and Cramer-type large-deviation asymptotics, for Pearson's product-moment sample correlation coefficient and a number of similar and more general statistics. Thus, the investigator aims to solve longstanding and difficult problems of probability theory and mathematical statistics. The first two of them concern some of the most important properties of such a classical and fundamental object as the Rademacher sums, whose distributions play the role of the extreme points of the set of the distributions of sums (and self-normalized sums) of any independent symmetric random variables. Extensions to the ``asymmetric'' case will also be considered. Closely related are other main objectives of the project, concerning limit theorems for self-normalized sums (or, equivalently, for Student's statistic).
The main impact will be in significantly better understanding of important properties of some of the most fundamental objects in probability theory and mathematical statistics. The successful completion of the project will also result in novel and important applications to such classical objects in statistics as Student's test and Pearson's correlation test, which are some of the very few hypotheses tests used most broadly in sciences and engineering. While there are great difficulties to overcome, it appears that the attainment of these objectives is within reach, given a number of advances already made by the investigator and his rather unique expertise in various areas of probability and statistics, as well as his demonstrated abilities to identify and solve difficult and longstanding problems and also to work effectively in a wide and highly diverse range of fields, including mechanical engineering, biology, operations research and combinatorics, and geometry and physics. Efforts will be made to disseminate results, not only via publication in wide-circulation journals, but also via news networks (stories on the investigator's work on evolution modeling and the Eiffel tower shape modeling have already been broadcast around the world by the United Press International and other news agencies). A number of graduate students will be involved into the project; efforts will be made to recruit from underrepresented minorities.
