Complete Convergence Of Moving Average Processes Of Dependent Random Variables

In this paper, we make a study on the complete convergence of moving average processes of dependent random variables. It is well known, that the convergence theorem is one of the most important results in probability theory.It is long time before people study the convergence theorem and limit theorem. In the early time, people studied the convergence theorem on i.i.d. real random variables. Recently,many people have studied the properties of negtively associated sequence(NA),linear negtive quadrant dependent sequence(LNQD) and dependent random variables. Furthermore, After the theory of real random variables get the perfect development, people begin to study probability theory in abstract space. Since 1953,people have studied the strong law of large numbers.the law of the iterated logarithm and complete convergence for B-valued random variables. In these years,some scholars try to study the qualities of complete convergence of moving average processes on the base of the convergence of randomvariables.,and they have obtained many important, results.In the 50s and in the 60s, after the limit theory about the sum of i.i.d. random variables get the perfect development, on the one hand because of the need of statistical problem, for example the sample is not independent, and some functions of the sample are not independent, on the other hand becauof the need of theory study and the other branch, such in Markov chain , random field theory (Ibrngimov- linnik 1971 ?9.1) and time series analysis and so on. Dependent random variables have a lot of important application, e.g. in reliability theory, Ising model of statistical mechanics, correspond theory, it is important to study them.In this paper,! try to use some results given by the formar scholars and study the complete convergence of moving average processes of dependence random variables.This paper has three parts. In the first section,we introduce the concepts of LNQD and 2 type of space.In the second section,we give and prove the complete convergence of moving average processes of LNQD sequence.In the third section.we discuss the complete convergence of moving average processes of dependent random variables.In this paper, we introduce the following definitions.lemmas and result.Definition 1.1 Let B be a separable Banach Space with norm ||.||. a real random variables (random elements) is called strictly stationary, if any positive integer m, n and 1 j1 < ... < jn.then (Xj1,... ,Xjn) = (Xj1+m, ... ,Xjn+m), a B valued sequence {Xn;n > 1} of strictly stationary random variables is called m dependent, where m is a nonnegative integer, if for any k 1, the two collections {X1, X2, ... , Xk|}and {Xk+m+1, Xk+m+2, ... } are independent.Definition 1.2 Two random variables X and Y are said to be NQD(negative quadrant dependent), iffor all x, y R.Two random variables X and Y are said to be NQD if and only if for any f,g, which are monotonically increasing functions, COV(f(X),g(Y)) 0.Random variable sequence {Xk, k 1} are said to be LNQD (linear negative quadrant dependent) ,if for any disjoint finite sub-sets A, B Nd and any positive real numbers rj, riXi andrjXj are NQD.Definition 1.3 Let B1,B2 be two separable Banach spaces ,and let L(B1, B2) be a, space composed by linear bounded transformation from B1 into B2. For L(B1, B2), if there be constant K > 0 such that for any finite independent elements X1 , ... , Xn B1), E, then we say is a 2 typetransformation.B are said to be a 2 type of space ,if identical transformation from B into itself is a 2 type transformation.Lemma 1.1 Let a, be an absolutely convergent seriesof real numbers with a = ai , b = |ai| . Let : [-b, b] - Rbe a function with the following properties: (1) is bounded, is continuous at a. (2) > 0 and C > 0,such that | (z)| C|x|, for all |x| Lemma 1.5 Let {Zk; 1 k n} be independent symmetricrandom variables and Tn = Zi. Then for every integer j 1,there exist positive constants Cj, Dj > 0 (Cj, Dj depending o...