Convergence Rate In Precise Asymptotics For The Law Of Large Numbers

Limit theory is one of the most important parts of probability theory.As the ba-sic theory of other branches of probability theory and mathematical statistics,it has an irreplaceable academic status.With the continuous development and expansion of limit theory,modern limit theory has emerged,and the research on precise asymptotic has become the focus and hot spot of scholars at home and abroad.At present,scholars at home and abroad have done much research on the precise asymptotics for the com-plete convergence,the law of iterated logarithm and the moment for the law of iterated logarithm,and have obtained mature theoretical results.However,the research on the precise asymptotics for the law of large numbers is quite few,so the study of the precise asymptotics for the law of large numbers in this paper has significance to the relevant theory.Through reviewing and summarizing the previous literatures,this paper mainly s-tudies the convergence rate in precise asymptotics for the law of large numbers.The theory of precise asymptotics has been paid more and more attention,not only in random variable sequences,but also in independent random variable,like ?-mixing sequences,p-mixing sequences and ?-mixing sequences.And many meaningful achievements have been made by scholars at home and abroad.However,most of these studies focus on the field of precise asymptotic,and there are few studies focusing on the convergence rate in precise asymptotics theory.In this paper,convergence rate in precise asymptotics for the law of large numbers is studied,and the following conclusions are obtained:Theorem Let {X,Xn,n ? 1} be a sequence of independent identically distributed random variables with EX = 0 and EX2 = ?2 ?(0,?),and set Sn =(?)Xi,n?1,let(?)and(?)under the moment condition EX2(log(1+|X|))1+?<?,we prove that(?)Inside,let L(n)be a slowly varying function satisfying property 1 and property 2 given by definition 2.2.1.For any given constant ? ?[0,?).n0 is a constant greater than or equal to 0.S is a constant greater than 0.This paper is divided into the following three parts:In the first chapter,we will mainly introduce the research background of the precise asymptotics and the convergence rate in the precise asymptotics,as well as the research results of scholars at home and abroad.And the structure arrangement of this paper will be given;The second chapter is the main content of this paper.Firstly,the preparatory knowl-edge will be introduced and the relevant lemmas needed for proof will be given.Secondly,the theorem of the convergence rate in precise asymptotics for the law of large numbers will be proved.Finally,the relevant corollaries will be given;The third chapter is a summary of this paper,and a new direction for convergence rate in the precise asymptotics for the law of large numbers as well as the prospect for the future research direction will be given.