Probability Limit Theorems and Statistical Applications

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Many of the best-known results in probability concern the asymptotic behavior of distributions; probability limit theory lies at the heart of probability and statistics. Probability limit theorems are concerned with rates of approximation of various observable processes by theoretically recognizable ones. For example, consider a random sample of n observations selected from a population with a finite variance. The central limit theorem states that, when the sample size n is sufficiently large, the sampling distribution of the sum of the n observations will be approximately normally distributed. Thus the central limit theorem allows us to make inferences on the mean of a distribution based on relatively large samples without having to know the exact form of the sampled population. There is a wide variety of statistical applications of probability limit theorems such as the analysis of large data sets, modelling traffic flow in communication networks, and providing a catalyst toward understanding and discussing the role of biostatistical research in relation to health services and decisions. The main focus of this research proposal will be on investigating the asymptotic behavior in statistical applications pertaining to hierarchical models, L-statistics, U-statistics, resampling methods, and the contemporary multivariate data analysis problems such as the largest entry of a sample correlation matrix, misspecified model, kernel estimator of the regression in a left truncation model, etc. A second focus relates to 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 or moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach-space-valued random processes. The goal are to develop new methods for proving limit theorems and to investigate statistical applications of these theorems. This provides a beautiful interplay between the theory and applications of Statistics, Probability, and Stochastic Processes. The results related to this proposal will be novel and significant insofar as they will extend, generalize, and refine earlier work in the literature.