【作者】 吕欣莉；

【导师】 王力；

【作者基本信息】 哈尔滨工业大学 ， 运筹学与控制论， 2009， 硕士

【摘要】 概率极限理论是概率统计学科中极为重要的基础理论。经典的极限理论,主要以独立随机变量为研究对象。随着独立随机变量和的经典极限理论获得较完善的发展,许多概率统计学家又投入到了各种混合序列收敛性质的研究中。这一方面是由于统计问题的需要,如样本并非独立,或者独立样本的一些函数也可能不是独立的;另一方面是来自理论研究及其他分支中出现相依性的要求,如马氏链、随机场理论及时间序列分析等,所以随机变量的相依性概念在概率论和数理统计的某些分支中被提出来了,而可交换随机变量和ρ混合序列就是相依随机变量中的两种很重要的类型。由于可交换随机变量的基本结构定理De Finetti定理——可交换随机变量无限序列以其尾σ?代数为条件是独立同分布的,因此可交换随机变量应该具有类似于独立同分布随机变量的性质。然而De Finetti定理仅对可交换无限列成立,存在着可交换随机变量的有限列,它不能嵌入到任何可交换随机变量无限列中去,所以必须寻找另外的办法解决可交换随机变量有限列的渐近性质的问题。ρ混合随机变量序列与通常的ρ混合随机变量序列有一定的类似,但并不完全相同,它们互不包含,在某些条件的要求上ρ混合随机变量序列比ρ混合随机变量序列要弱的多,应用的领域也更加的宽泛。本文是在已有结果和方法的基础上来研究两类应用较为广泛的相依随机变量,即可交换随机变量及ρ混合序列的有关的一些问题,主要对可交换随机变量及ρ混合序列的极限性质进行了如下的讨论:首先,讨论了可交换随机变量部分和的几乎处处收敛性。其次,通过巧妙地截尾建立对证明起关键作用的集合包含关系式,将独立情形下的Katz和Baum定理推广到了可交换随机变量。最后,讨论了ρ混合序列加权和的收敛性,将独立同分布情形下的Thrum和Srout定理推广到了可交换ρ混合序列。更多还原

【Abstract】 Probability limit theory is important basic theory of the science of probability and statistics. Limit theory mainly study independent random variables. About Limit theory of random variables, its convergence properties have drawn many attentions from scholars for its extensive application. On the one hand, it is investigated quite perfectly but in many practical problems, samples are not independent, or the function of independent sample is not independent; On the other hand, the concept of dependent random variables in probability and statistics is mentioned, such as Markov chains, random environments, time series Exchangeable random variables andρ-mixing sequence are a major type of dependent random variable.As the fundamental structure theorem of infinite exchangeable random variables sequences, the De finetti’s theorem states that infinite exchangeable random variables sequences is independent and identically distributed with the condition of the tailσ-algebra. So some results about independent identically distributed random variables are similar to exchangeable random variables. As the fundamental structure theorem of infinite exchangeable random variables sequences, the De finetti’s theorem does not work on finite exchangeable random variables sequences, it is therefore necessary to find other techniques to solve the approximate behavior problems of finite exchangeable random variables sequences. By using reverse martingale approach, some scholars have given some results. In this paper we do some researches about the similarity and difference of identically distributed random variables and exchangeable random variables sequences, mainly discuss the limit theory of exchangeable random variables andρ-mixing sequence.Firstly, we discuss almost surely convergence about sums of exchangeable random variables.Secondly, we extend set methodology by using censore. We extend the Baum and Katz theorem in the condition of independent, and obtain the complete convergence theorem of expression of Baum and Katz theorem in the case of exchangeable random variables.Finally, we discuss sure convergence for weighted sums ofρ-mixing sequences. We extend the Thrum and Srout theorem in the condition of independent, obtain a result of convergence about weighted sum forρ-mixing sequence about almost surely convergence and complete convergence.更多还原