Some Researches On Almost Sure Central Limit Theorems And Precise Asymptotics

The classical central limit theorem has served as a basis for large group of investigations of fundamental significance both in the theory of probability and in its numerous applications to the statistics, natural sciences, engineering and economics. Its methods and results continue to have great influence on other fields of probability theory, mathematical statistics, and their applications.In 1988, a new chapter of limit theorem has been discovered, that is called almost sure central limit theorem (ASCLT). It is also called as pointwise central limit theorem.In Chapter 1 we are mainly concerned with almost sure limit theorems for the nonstationary random variables. We prove almost sure limit theorems for partial sums and maxima of nonstationary normal random variables. Meanwhile, a general pattern of almost sure central limit theorem is also considered.As early as more than sixty years ago, in the study on the strong convergence of random variables, Hsu and Robbins (1947) and Erdos (1949,1950) brought forward the conception of complete convergence. Later, Spitzer (1956) improved their conclusions further. In 1960 s, Baum and Katz applied Erdos and Spitzer s method and came up with the well-known conclusion. Later, based on the above ultimateness, various conclusions have been established. Recently, Gut and Spataru (2000a) discussed precise asymptotics in the Baum-Katz and Davis laws of large numbers.Chapter 2 mainly focus on some kinds of precise asymptotics for self-normalized sums. We derive the precise asymptotic in complete moment convergence of self-normalized sums for multidimensionally indexed random variables.Chapter 3 develope the topic of precise asymptotics on some stochastic processes, such as counting process and uniformly empirical process and derive some general result.