摘要
本文分为五章，讨论了在相依变量的情形下的完全收敛性和重对数律。
第一章，讨论了不同分布NA随机变量序列加权和的完全收敛性，获得了较已有结果更为一般的完全收敛性，并得到了完全收敛速度与矩条件之间的等价关系。
第二章，讨论了ρ~--混合随机场的部分和的完全收敛性。在一些适当的条件下，获得了较为一般的ρ~--混合随机场的部分和的完全收敛性定理，并得到了完全收敛速度与矩条件之间的等价关系。所得的结果推广了ρ~*-混合随机场和负相伴序列的相应的结果。且将Hsu-Robbins型定理推广到ρ~*-混混合随机场的情形。定理的证明基于Rosenthal型最大值不等式，Rosenthal型不等式，几个引理及缓变函数的性质。
第三章，讨论了不同分布ρ~--混合随机场的部分和的完全收敛性，建立了一个定理，此结果的获得推广了ρ~*-混合随机场和NA序列的相应的结果。
第四章，讨论了ρ-混合序列加权和的完全收敛性，并将此结果应用于线性回归模型参数β的最小二乘估计及非参数回归模型g的权函数估计中，所得的结果改进了已有的相应的结果。
第五章，设｛X_n，n≥1｝是同分布ρ-混合序列，其分布属于特征指数为α(0＜α＜2)的非退化稳定分布的正则吸引场，证明了依概率1有
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 have
limsup a.s.
引文
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1．蔡光辉．ρ~--混合随机场的完全收敛性，浙江大学学报(理学版)，已录用．
2．蔡光辉．不同分布NA序列加权和的完全收敛性．应用数学，2002，15(3)，106—110．
3．蔡光辉，张立新．不同分布ρ~--混合随机场的完全收敛性．高校应用数学学报，已投稿．
4. Cai, G. H., Complete convergence of moving average processes under negatively associated assumptions, Math. Appl.(in chinese), Accepted.