General Limit Theorems in Probability with Applications to Statistics

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My research proposal is devoted to general limit theorems in probability and asymptotic statistics and in their applications to a wide variety of problems. We live in a random world. Probability theory is the branch of mathematics concerned with analysis of random phenomena. Limit theory lies at the heart of probability and statistics and plays a central stage in almost every branch of science or social science including weather forecasting, psephology, etc. Results such as the law of large numbers, the central limit theorem, and the law of the iterated logarithm for independent random variables have given shape to modern probability theory. They have been extended and generalized in many directions, among others, to more general random processes and random measures, and they have become the bases of asymptotic statistics. Asymptotic statistics or large sample theory, is a generic framework for the assessment of properties of estimators and statistical tests. Within this framework, it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as the sample size n tends to infinity. The first focus of this research proposal relates to my my long-standing research interest in almost sure and weak convergence of random processes, especially in the law of the iterated logarithm, the laws of large numbers, central limit theorems, probabilities of large and moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach space-valued random processes. The goal is to study refinements of the classical limit results and to develop some new methods for proving almost sure and weak convergence of random processes and to continue my previous research work, i.e., to use modern random process techniques in probability and to develop some new probability inequalities for random processes in order to find conditions under which almost sure and weak convergence holds for random processes and to investigate statistical applications of such convergence. A second focus will be on investigating the asymptotic behavior in statistical applications pertaining to hierarchical models, L-statistics, U-statistics, resampling methods, and high dimensional data analysis problems such as the largest entry of a sample correlation matrix, estimation of conditional density and mode with truncated and censored data, etc. For example, motivated by a statistical hypothesis testing problem, asymptotic behavior of the largest entry of a sample correlation matrix has been studied extensively in recent years including my three refereed journal articles: 1. The Annals of Applied Probability, Vol. 16, 423-447, 2006 (with A. Rosalsky), 2. Probability Theory and Related Fields, Vol. 148, 5-35, 2010 (with W. Liu and A. Rosalsky), and 3. Journal of Multivariate Analysis, Vol. 111, 256-270, 2012 (with Y. Qi and A. Rosalsky). The successful completion of my proposed work would be an important step in increasing our understanding of the asymptotic behavior of the largest entry of a sample correlation matrix in very general and applicable situations. The results related to this proposal will be novel and significant insofar as they will extend, generalize, and refine earlier work in the literature. All results will be formalized in papers for publication in major academic journals.