【作者】 宋家乐；

【导师】 王力；

【作者基本信息】 哈尔滨工业大学 ， 概率论与数理统计， 2009， 硕士

【摘要】 概率极限理论不仅是概率论的主要分支之一,而且也是概率论其它分支以及数理统计的重要理论基础。前苏联著名的概率学家Kolmogorov曾说过:“概率论的价值只有通过极限定理才能被揭示,没有极限定理就不可能去理解概率论中基本概念的真正含义。”经典的中心极限定理是概率论的重要基础,它已被广泛应用于统计、自然科学、工程学和经济学等领域,其方法和结果还将继续对概率论的其它分支,数理统计产生巨大影响。而几乎处处中心极限定理(Almost sure central limit theorem,简称ASCLT)则是近二十年来概率极限理论研究的重要方向之一。自从Brosamler和Schatte在1988年首先提出了独立同分布随机变量序列的几乎处处中心极限定理以来,Berkes、邵启满、Multa等都对几乎处处中心极限定理做过研究。本论文将在前人给出的一些随机变量序列几乎处处中心极限定理相应结果的基础之上,对不一定同分布的随机变量序列部分和乘积的几乎处处中心极限定理进行了讨论。首先,我们介绍了有关几乎处处中心极限定理的研究进展,给出了一些相依随机变量序列的几乎处处中心极限定理。其次,我们在不一定同分布的条件下,讨论了鞅差序列部分和乘积的渐进正态性,并且在此基础之上,证明了强混合鞅差序列部分和乘积的几乎处处中心极限定理。再次,我们还在适当的条件下,证明了不一定同分布的φ、ρ-混合序列部分和乘积的渐近正态性。最后,我们介绍几乎处处中心极限定理在统计学中的研究进展,介绍了关于顺序统计量、U -统计量的几乎处处中心极限定理的一些结果。更多还原

【Abstract】 Probability limit theory is not only one of the main branches of probability theory, but also is an important theoretical foundation of other fields of probability theory and mathematical statistics. The famous Soviet probabilist Kolmogorov said:“only probability limit theory can reveal the epistemological value of probability. Without the theory,you could not understand the real meaning of the fundamental conceptions in probability.”The classical central limit theorem is an essential foundation of probability theory, which is extensively applied to statistics, nature sciences,engineering and economics, etc. Its methods and results will continue to have great influence on other fields of probability theory, mathematical statistics, and their applications. The almost sure central limit theorem have become an important field of the study of probability limit theory in recent decades.Ever since Brosamler and Schatte proposed the almost sure central limit theorem of i.i.d. in 1988, a lot of people have studied the theory. In this paper, based on the existing conclusions, we discuss the ASCLT from the following three aspects:Firstly, we give some introductions about the ASCLT of some dependent random variables.Secondly, under the assumption of nonidentical distribution, we prove the asymptotic distribution of the product of martingale difference sequence sums. Based on that conclusion, we prove the almost sure central limit theorem of the product of martingale difference sequence sums underα-mixing. Besides, under the assumption of nonidentical distribution, we also prove the asymptotic distribution of the product ofφ、ρ-mixing partial sums.Finally, we give some introductions about the ASCLT in Statistics.更多还原