The Dependent Variables Complete Convergence With Re-logarithmic Law

This dissertation consists of five chapters, in which we discuss the complete convergence and the iterated logarithm under dependent random variables.In chapter one, we investigate the complete convergence for sums of non-indentically distributed NA random variable sequences. We obtain a more general complete convergence than the result appeared in literature. And we obtain the equivelent relationship between rates of complete convergence and moment condition.In chapter two, the complete convergence for sums of ρ--mixing random fields are discussed. The general theorems of the complete convergences for sums of ρ- - mixing random fields are obtained under some suitable conditions. And the equivalent relationship between rates of complete convergence and moment condition is obtained. The results obtained extend the relevant results for ρ* -mixing random fields and nagatively associated sequences. And Hsu-Robbins type theorem is generalized to the situation of ρ- -mixing random fields. The proof is based on Rosenthal type maximal inequality, Rosenthal type inequality, several lemmas and properties of slowly varying function.In chapter three, we investigate the complete convergences for sums of non-indentically distributed ρ--mixing random fields. We obtain a theorem, which extend those for ρ*-mixing random fields and negatively associated sequences.In chapter four, we investigate the complete convergence for weighted sums of ρ-mixing sequences. We apply results to the least square estimations in linear regression models and weighted function estimates g in nonparametric regression. The results improve the results appeared in literature.In chapter five, let {Xt,i 1} be ρ-mixing sequences with identical distributions, which belong to domain of normal attraction with non-generational and stable distribution. With probability one, we havelimsup a.s.