Some Study Of Limit Properties And Complete Convergence For Series Of Dependent Random Variables

The strong deviation theorems,which are also called the small deviation theorems(i.e.,the strong limit theorems represented by inequalities),are new type theorems established by using the notion of the likelihood ratio.In chapter 2 the strong deviation theorems for functions of series of discrete random variables are given. Here we construct a moment generating function and a convergent martingale .In the main time, we use the truncation method and the properties of monotone functions . In recent ten years,Prof. Liu and his cooperaters have done a lot of work about the strong limit theorems of fair gambling system.In chapter 3 we make a convergent martingale and conclude the strong limit theorem of bounded gambling system,which is further study on the basis of previous work. In chapter 4,we obtain the strong limit theorems about series of random variables which are in the a-th(l < α≤ 2) step smooth space.Here we use the truncation method and the properties of monotone functions and then make a convergent martingale.In 1947,Hus and Robbin defined the complete convergence,which is stronger than almost everywhere convergence. In chapter 5 we prove the complete convergence of series of dependent random variables by making a convergent martingale .And the martingale is bounded.This paper contains six chapters.In chapter l,we introduce the relative background on this paper and give some simple expressions of the work which have been studied. Chapter 2-5 are the main parts of this paper.In chapter 6,the main results are given.