Asymptotic Behavior For Robbins-Monro Process

In this thesis,we study the asymptotic behavior for Robbins-Monro procedure X_n+1=X_n-an^-1Y_n with some fixed number a>0,including mainly the convergence and convergence rate,and we establish the moderate deviation principle of the process {X_n}.In Chapter 1,we first introduce the background of the Robbins-Monro process and present some well known results in literature.Then we list the basic concepts and properties of the process and educe the research under my consideration on convergence rate and moderate deviation principle of the Robbins-Monro process.In Chapter 2,we firstly list some basic assumptions on the function M?x?and errors{V_n}.Then,the exponential inequality and its proof are given under the condition of the basic assumptions.The essence of the exponential inequality is that the convergence rate to zero of Robbins-Monro iterationxn is in exponent changing.In Chapter 3,we present our main researclh results on the moderate deviation prin-ciple of the Robbins-Monro procedure {X_n} in this chapter.Specially,we strengthen the exponential integrable condition of the errors to the boundness,present the moderate de-viation principle of the process{X_n}:lim_n??1/bn²log P(h_n|X_n+1|?rb_n =-r²/2?²,where ?²=Var?V₁?,r>0 and h_n=(a²?_m=1ⁿ m²?_mn?a?₁?)^-1/2 and then give the detailed derivation process.