Probability Asymptotic Theorems and Their Applications

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This research proposal is devoted to topics related to general asymptotic theorems in probability and statistics, and their applications to a wide variety of problems. Probability asymptotic 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 basis of asymptotic statistics or large sample theory which 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 laws of large numbers, central limit theorems, the law of the iterated logarithm, 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, to develop some new methods and to establish some new probability inequalities for proving almost sure and weak convergence of random processes. I also aim to continue my previous research, using modern random process techniques in probability 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 of this research proposal will be on investigating the asymptotic behaviour in statistical applications pertaining to high dimensional data analysis problems, hierarchical models, resampling methods, random matrix theory, etc. Therefore I propose to study the asymptotic theorems for a) largest entries of bootstrapped sample correlation matrices, b) spectral radii for product of matrices from the spherical ensemble, c) hierarchical models, etc. The successful completion of my proposed work would be an important step in increasing our understanding of the asymptotic behaviour of random processes in very general and applicable situations since the conclusions of an asymptotic theorem often supplement the conclusions which can be obtained by numerical methods. The results to be obtained in this proposal will be novel and significant in that they will extend, generalize, and refine earlier work in the literature. All results will be formalized in papers submitted for publication in high-impact academic journals.