Precise Asymptotics In The Law Of Large Numbers Of Moving-average Processes

Theory of Probability is a science of quantitatively studying regularity of random phenomena , which is extensively applied in natural science, technological science, social science and managerial science etc. Hence, it has been developing rapidly since 1930'sand many new branches have emerged from time to time. Probability Limit Theory is one of the branches and also an important theoretical basis of science of Probability and Statistics.Convergence is an important conception in Probability Limit Theory. There are a series of convergence conception such as convergence in probability, convergence in distribution, almost sure convergence, while Hsu and Robbins (1947) were the first one to give the definition of complete convergence, then they and Erdos (1949,1950)established : for allε> 0if and only if EX = 0, EX~2 <∞. In the 60's of the 20th century, Katz (1963)and Baum and Katz (1965) extended their results: Let 1≤p < 2, r≥p, thenif and only if EX = 0, E|X|~T <∞, r≥1.Later, based on the above ultimateness, various conclusions have been developed. Recently, Gut and Spataru (2000a) discussed precise asymptotic in the law of logarithm of i.i.d random variable sequence, they established the result as following: let EX = 0, EX~2 <∞, then for all 0≤ε≤1,As is known to all, everything has correlations between one another in the world. If we can properly describe these correlations by mathematics, we can analyze subjects accurately by the precise tool-mathematics. Hence one can see that, the study on dependent random variables has momentous significance. In Chapter One, we focus on the study of precise limit theorems of negatively associated (NA) sequences. The concept of negatively associated was introduced by Alam and Saxema (1981) and Joag-Dev and Proschan (1983). Since this concept has been widely applied in some practical models (for example, reliance theory, filtering theory and multivariate analysis etc.), it has attracted close attentions from many scholars.we apply purely algebraic decomposition for moving-average processes, it turns out to be a useful device in reducing moving-average asymptotic to known theorems for i.i.d, negatively associated and stationary sequences. We got the result as following:Let {ε_i; -∞< i <∞} be NA and stationary r.v sequences. Eε_i = 0, Varε_i < +∞. {a_j} is real coefficients which satisfy the following conditions: (?) , then for all p > 0, we got the result:Gut and Sp(?)taru (2000) discussed Precise Asymptotic in the law of large number for Baum-Katz, whenε→0. and they established the below result: let {X_k; k≥1} be i.i.d random variable sequences EX_i = 0,0 < EX_i =γ~2 <∞, then for all 1≤p < r, we have:Z is a normal r.v., and EZ = 0, VarZ -γ~2.Y.X.Li(2006) extend the result to moving-average processes where innovation term is NA and stationary. And in our second chapter, under some conditions, we extend the result above as follow:Theorem : {a_i; -∞< i <∞} is real coefficients, satisfying such conditions: (?). meanwhile (?)(x) satisfy condition: (?)(x) is positive function, and (?)(x) (?)∞, and (?)'(x)is also positive; (?)'(x) is monotonically nondecreasing function, Then for all v > 1/2 we have:...