Non-uniform Berry-Esseen Bounds For Unbounded Exchangeable Pairs

Limit theorems and the convergence rate are one of the most attractive fields for probabilists.In normal approximation,the error between the random vari-ables {Wn} under study and the standard normal distribution function ?(z)(?) is called the(uniform)Berry-Esseen bound.When the bound also depends on"z",it is called the non-uniform Berry-Esseen bound.The characteristic function method has given the uniform and non-uniform Berry-Esseen bounds for sums of independent random variables[1][2][13].However,it is difficult to handle depen-dence by characteristic functions.To overcome this setback,Charles Stein[24]invented a new method in 1972,which has been called Stein's method-Consider the following Stein equation:fz'(x)-xfz(x)=I(x?z)-F(z),x ?R.Applying this to a random variable W a.nd taking expectation on both sides,we can calculate the expectation on the left side to estimate the error.Stein's method has proved to be powerful in many fields,including the non-uniform approxima-tion.By Stein's method,Chen and Shao[9]gave the non-uniform Berry-Esseen bounds for sums of independent random variables and later Chen and Shao[10]also established non-uniform bounds for dependent random variables.The main technique in their papers is the concentration inequality approach,which also has close relation to the exchangeable pairs introduced in the following.Exchangeable pairs are another significant conception in Stein's method.A pair(W,W')is called an exchangeable pair if(W,W')and(W',W)have the same distribution.The previous research focused on the bounded situation where|W-W'|<?.But when |W-W'| is unbounded,there was no very good result.Recently,Shao and Zhang[22]made a breakthrough for unbounded exchangeable pairs.They established easy and applicable bounds for unbounded exchangeable pairs in normal and non-normal approximation,and achieved good rates in many important examples.Their work does not rely on the concentration inequality approach.In this thesis,a new technique is introduced to obtain non-uniform Berry-Esseen bounds for normal and non-normal approximations by unbounded ex-changeable pairs.This technique does not rely on the concentration inequalities developed by Chen and Shao[9,10]and can be applied to the quadratic forms,the general Curie-Weiss model and an independence test.In particular,our non-uniform result is under 6th moment condition,while the uniform result in Chen and Shao[11]requires 24th moment condition.This thesis is organized as follows:in the first chapter,we introduce the history and related results;in the second chapter,we give our main result;in the third chapter,we give the proof of the main result;in the forth chapter,we show the applications,which are quadratic forms,the general Curie-Weiss model,and an independence test;in the fifth chapter,we conclude our contributions,reflect on the possibility of improvement and make some prospects.