Precise Asymptotics Of Random Variables

Theory of Probability is a science of quantitatively studying regularity of random phenomena, which is extensively applied in natural science, technological science, social science and managerial science etc. Hence, it has been developing rapidly since 1930's and many new branches have emerged from time to time. Limit Theory is one of the branches and also an important theoretical basis of science of Probability and Statistics. As stated in the classical book "Limit distributions for sums of independent random variables"(1949) by B.V.Gendenko and A.N.Kolmogrov, "The epistemological value of the theory of probability is revealed only by limit theorems. Without limit theorems it is impossible to understand the real content of the primary concept of all our sciences-the concept of probability."Limit theory also develope quickly. Traditional research contents have CLT, LLN, LIL and complete convergence, and also include precise asymptotics which develope on the base of complete convergence. Recently in Jiang(2004) and Li (2005) they consider a moment type precise asymptotics of various dependent sequence and sequence producted by various process respective. The main contribution of this paper is to board this moment type precise asymptotics to bootstrap and self-normalized partial sums which is very useful in statistics. And there is also improvement of method.The first chapter we introduce the developement of classical limit theory, and give the backup of birth of this moment type precise asymptotics.The content of the second chapter is about precise asymptotics of bootstrap means. The first section we introduce the bootstrap and list out some recently results about it, the second section is to give the moment type precise asymptotics of bootstrap means, and in the method of prove, we abandon out the Berry-Esseen inequality which is used in Jiang(2004) and Li(2005), the merit of doing this is that we can reduce the moment conditions in theory.The third chapter we study the self-normalized partial sums, in first section we introduce the merit of self-normalized partial sums in statistic application and also include some develope histroy and rencently results, the second section we also consider its moment type precise asymptotics, and include law of log and iterad log.