A Class Of Random Deviation Theorems And The Approach Of Laplace Transform

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 1989,Professor Lin Wen first applied an analytic technique in the study of a class of strong deviation theorems for a sequence of random variables[34]. Later,Professor Liu Wen studied the sequences of discrete random variables by combining the analytic technique with the approach of generating function and moment generating function and obtained a class of deviation theorems[10]~[121. In this paper, by means of the notion of likelihood ratio and log likelihood ratio the limit properties of the sequences of dependent continuous random variables are studied,and a class of strong limit theorems represented by inequalities are obtained.In the proof an approach of applying the tool of Laplace transform to the study of strong limit theorem is proposed.This paper contains five chapters. In chapter l,we introduce the relative background on this paper and give some simple expressions of the work which have been studied. In chapter 2, in virtue of the notion of likelihood ratio the limit properties of the sequences of dependent nonnegative continuous random variables are studied, and a class of strong limit theorems represented by inequalities are obtained.The bounds given by these theorems depend on positive constant c. In chapter 3,by means of the notion of log likelihood ratio,a kind random strong deviation theorem are obtained, and the bounds given by these theorems depend on r( ). In the proof,the notion of tail probability and its Laplace transform is proposed. In chapter 4,The purpose of this chapter is to establish a kind of strong deviation theorems of functional for the sequences of arbitrary continuous random variables, by using the conception of log likelihood ratio,and extend the strong deviation theorems on the differential entropy for dependent arbitrary continuous information sources on the the probability space ( , .F,P). In chapter 5, the main results are given in this paper.