| In the 1980's, the concept of negatively associated random variables, which include the concept of independent random variables, was introduced by Alam and Saxena(1981), Blok, Savits and Shaked(1982), Joag-Dev and Proschan(1983) and Joag-Dev(1983).Definition A A finite family of random variables {Xi, 1 < i < n) are said to be negatively associated (NA in short) if for every pair of disjoint subsets whenever f1 and f2 are real coordinatewise increasing and the covariance exists. An infinite family is negatively associated if every finite subfamily is negatively associated.Since the concept of NA random variables has a lot of applications in multivariate statistical analysis, reliability theory, percolation theory, many engineering problems and risk analysis theory, various aspects of NA random variables are significant and have been investigated by a lot of scholars.In recent ten years after that the definition of NA random variables has been introduced, the research development of them was rather slow relative to that of other associated random variables, e.g. mixing random variebles and positively associated random variebles. Until 1992, in the paper of Mat-ula(1992), Polish scholar, Matula established the Kolmogrov upper bound inequality and the three series theorem for the NA random variables, and gavethe Kolmogrov strong law of large number for the identical distribution NA random variables which was completely same as the independent situation. After that, many scholars investigated some aspects of the limit theorems of NA random variables, statistics and their applications, and obtained many important results, such as Su Chun and Wang Yuebao(1998) established Marcinkiewicz strong law of large number, three series theorem and uniform convergence for NA random variables with identical distribution which was same as in-dipendent situation; Su Chun(1996) established Hsu-Robbins theorem for NA random variables; Pan and Liu(l998) established Esseen inequality for NA random variables; Su Chun, Zhao Lincheng and Wang Yuebao(1996) established the weak invariability principle; Shao and Su(1999) established the bounded law of the iterated logarithm for NA random variables with stationary distribution; Hu Yijun(1998a) established large deviation principle for NA random variables with stationary distribution.In this article, we investigate some problems of the limit theorems of the NA random variables, research on some aspects of the almost sure central limit theorem, moderate deviation principles and large deviation principles, and obtain some rather diverting results.1. On condition that the second order moment exists, we obtain the almost sure central limit theorems for NA random variables with stationary distribution.2. We obtain the moderate deviation principles for NA random variables with stationary distribution under a weak condition.3. We obtain the moderate deviation principle and large deviation prin-ciple for the moving average process of real random variables with stationary distribution, then the ones for NA random variables is obtained.Main ResultAs the first part, of this article, we consider the problem about the almost sure central limit of NA random variables. Almost sure central limit theorem (ASCLT in short) is a pop topic of the probability research in recent years. Suppose {Xk,k > 1} is a real random variable sequence on Defines the convert-line process. [nt]here we suppose that sn(0,w) = 0. ASCLT problems mainly focuses on conditions, on whichand (2)hold. Here x occured in (1) is a single-point measure over R, x occured in (2) is a single-point measure on the continuous function space C[0. 1]. N(0, 1) is standard normal distribution, and H+ is the standard Brown movement on [0,1].In this part, we prove that NA random variables with stationary distribution satisfy (2) on condition that the second order moment exists. At the sametime, we investigate the ASCLT for LNQD random variables in this part, and prove that (1) holds on condition that the...
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