Sequence Of Random Variables Convergence

This paper discusses from the point of complete convergence the convergence of random sequences which include both i.i.d. and dependent case, such as NA sequences and -mixing sequences. As is well known that i.i.d random sequences are special cases of NA sequences or p- mixing sequences which have not properties as good as i.i.d random sequences. But all things in our world are connected one with another. So, it is very meaningful to study dependent random sequences for the reason that we are more possible to encounter dependent cases than i.i.d cases.The concept of complete convergence was introduced by P.L.Hsu and Robbins [11] in 1947: for arbitrary > 0 and i.i.d random variables with mean zero and variance 1, we haveThis conclusion strengthened classical strong law of large numbers in the direction of Borel-Cantelli lemma. Later, Erdos [12] sharpened the above result, proving that those conditions put on moments are sufficient and necessary. Baum and Katz [13] generalized these results to the cases of fractional and high order moments. From then on, many scholars have studied the convergence of random variables form the point of complete convergence. Y.S.Chow and T.L.Lai [3] proved the following:where Ap, > 0, Bp, > 0 are absolute constants. Pruss [1] discussed the same problem, proving that for i.i.d random sequences {X,Xn : n 1} with mean zero, the following holds true:where 0 < C1,C2 < are absolute constants.This result strengthened the results of Y.S.Chow without discussing the cases of fractional and high order cases. It is known that we may encounter more often such cases than one order ones. Our results in chapter one generalize the results and considered more general cases, that is, we have the following result:Proposition 1: let {X,Xn : n > 1} are i.i.d. random variables with mean zero, p > 2, A > 0. Define Y = XI. If 1, thenIt > 1/2,thenwhere 2 = EX2, then when A is large enough, In chapter two we discuss the complete convergence of NA random sequences. Let be a finite index set. A set of random variables, say, {Xi;i }, is called negatively associated, if for arbitrary A and B which are two disjoint subsets of ,where f1 and f2 are two coordinatewise monotone functions which assure the above two covariances exist; A random sequence {Xi,i 1} is called negatively associated, if for arbitrary natural numbers n 2, {X1,X2,... Xn} are all negatively associated. In many practical models such as reliability theory, filtering theory and multivariate analysis, NA sequence has so many useful applications that it attracts many scholars working on it, and up to now, there have been many results. Recently, Wang Yuebao [4] proved the following equivalent conditions for the complete convergence of i.i.d random sequences:Assume r > 1, {Xnk; k Z,n N} be independent, real valued random sequences. Thenhold if and only ifandwhen {Xnk} are nonsymmetric.We generalize the result to NA sequences, and also, a main result in [4] to NA sequence without changing the original conditions much. Our result is as follows:Proposition 2: Let {Xnk : n 1,1 k n} be NA sequence, l(x) be a slow-varying function when x - , r > 1. Define the following symbols:IfthenOn the contrary, if we have (0.12) and (0.10), then we also have (0.11).The third chapter discusses the convergence rates of NA sequences in LIL. LIL is a very precise phenomena whose rates of convergence have attracted many scholars' attention. [7] and [8] discussed the rate of convergence in LIL of i.i.d. random sequences; [9] discussed the convergence rates in the law of logarithm of NA sequences. [14] discussed the convergence rates in LIL of i.i.d. B-valued random sequences. [10] discussed the general forms of the rates of convergence in LIL of B-valued random sequences.[7] proved that for real valued i.i.d. random sequence (X,Xn,n 1}, if EX = 0,EX2 = 1, and EX2 log log |X| < , then for Me > 0,[8] proved that for real valued i.i.d. random sequence {X,Xn,n 1}, if EX = 0...