【作者】 鄢寒；

【导师】 吴群英；

【作者基本信息】 桂林理工大学 ， 统计学， 2009， 硕士

【摘要】 概率论是从数量上研究随机现象的规律性的学科.它在自然科学、技术科学、社会科学和管理科学中都有着广泛的应用,因此从上世纪三十年代以来,发展甚为迅速,而且不断有新的分支学科涌现.概率极限理论就是其主要分支之一,也是概率统计学科中的极为重要的理论基础.而近四十年来,其中的完全收敛性和强收敛性已经成为当前概率极限理论研究中的最重要的热门方向之一.本文也就此方面着手,研究了两类重要的随机变量序列的强极限定理,并得到了一些精确的强极限结果.众所周知,现实生活中所发生的事情大多并不是互不相干,而是彼此之间具有某种联系的.正确地用数学方法来描述这种相关性,就可以用数学——这一精确的工具来对事物进行精确的分析.由此可见,研究非独立的随机变量序列有着十分深刻的理论和实际意义.其实,关于相依随机变量的极限性质的研究可以追溯到二十世纪二、三十年代,当时就有Bernstein （1927）、Hopf （1937）和Robbins （1943）等学者相继对其进行研究.一直到现在,仍有新的相依变量类型及其结果层出不穷.而在本文中,我们就其中两种较为常见的随机变量进行了一些方面的讨论.内容主要包括如下三章:第一章研究了两两NQD序列的收敛性质.主要讨论了两两NQD阵列行和的弱大数律、L p收敛性和完全收敛性,在{X_nk;1≤k≤k_n↑∞, n≥1}是Cesàro一致可积的相关条件下,获得了两两NQD阵列行和的弱大数律、L p收敛性和完全收敛性定理,将独立阵列行和的相关极限定理推广到了两两NQD阵列行和的情形.第二章和第三章讨论了ρ^-混合序列的收敛性质.第二章我们讨论了ρ^-混合序列的完全收敛性和Marcinkiewicz强大数律,获得了与独立情形完全一样的Baum和Katz定理和Marcinkiewicz强大数律.第三章讨论了ρ^-混合序列加权和的完全收敛性和强收敛性,所得结果推广了Thrum和Stout定理.更多还原

【Abstract】 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 l930’s and 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. During the past forty years, complete convergence and strong convergence have become the most important and popular orientations of the current study of Probability Limit Theory. Starting with the above mentioned points, we obtain some limit theorems of two types of important random variables, and draw some precise results in this thesis.As is known to all, everything has correlations between one another if 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 fact, the study on the limit properties of dependent random variables may be dated to 1920’s and 1930’s, at that time ,scholars such as Bernstein（1927）, Hopf（1937）, Robbins（1948） had carried on studies on this topic. Till now, new kinds of dependent random variables and their corresponding conclusions have emerged in a endless stream. This article is deemed to take two common kinds of dependent random variables. It is divided into three chapters as follows:In Chapter one, some limit properties of Pairwise NQD sequences have been discussed. The week law of large numbers, L pconvergence and complete convergence of the maximum of sums of pairwise NQD random matrix sequences are discussed. Under the condition that the {X_nk;1≤k≤k_n↑∞, n≥1} is Cesàro uniformly integrable, the authors are able to give the week law of large numbers, L pconvergence and complete convergence of the maximum of sums of pairwise NQD random matrix sequences, which generalize the corresponding limit results for independent random matrix sequences to pairwise NQD random matrix sequences.In Chapter two and Chapter three, some limit properties ofρ^- -mixing random sequences have been discussed. In Chapter two, the complete convergence and Marcinkiewicz strong laws forρ^- -mixing random sequences are discussed. As a result, Baum and Katz complete convergence theorem and Marcinkiewicz strong laws are extended to the case ofρ^- -mixing random sequences. In Chapter three, we establish some sufficient conditions of the complete convergence and strong convergence for weighted sums ofρ^- -mixing random sequences.The results obtained extend the theorem of Thrum and Stout.更多还原