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.
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.
引文
[1] Stout W F. Almost Sure Convergence[M]. New York: Academic Press, 1974
[2] Valentin V.Petrov. Limit Theorems of Probability Theory Sequences of Independent Random Variables[M]. OXFORD, Clarendon Press, 1995
[3]陆传荣,林正炎.混合相依变量的极限理论[M].北京:科学出版社, 1997
[4] Lehmann E. L. Some concepts of dependence[J]. Ann .Math. Statist., 1966, 43 :1137-1153
[5] Joag-Dev K., Proscha F. Negative Association of Random Variables with Applications[J]. Ann. Statist., 1983,11: 268-295
[6] Matula P. A Note on the Almost Sure Convergence of Sums of Negatively Dependent Random Variables[J]. Statistics and Probability Letters, 1992, 15: 209-213
[7] Wang Yuebao, Su Chun, Liu Xuguo. On some properties and its applications for pairwise NQD sequences[J]. Acta Mathematicae Applicatae Sinica, 1998, 21(3): 78-83
[8]吴群英.两两NQD列的收敛性质[J].数学学报, 2002, 45(3): 617-624
[9]张立新,王江峰.两两NQD列的完全收敛性的一个注记[J].高校应用数学学报A辑, 2004, 19(2): 203-208
[10]吴群英.两两NQD列的广义Jamison型加权和的强收敛性[J].数学研究, 2001, 34(4): 386-393
[11]王岳宝,严继高,成凤旸,蔡新中.关于不同分布两两NQD列的Jamison型加权乘积的强稳定性[J].数学年刊. 1998,(3): 389-395
[12] Chandra T K. Uniform integrability in the Cesàro sense and the week law of large numbers[J]. Sankhya: The Indian Journal of statistics, 1989, 51(serise A): 309-317.
[13]吴群英.混合序列的概率极限理论[M].北京:科学出版社. 2006
[14] Baum L E, Katz M. Convergence rates in the lay of large numbers[J] .Trars Amer Math Soc., 1965, 120: 108-123
[15]白志东,苏淳.关于独立和的完全收敛性[J].中国科学,A辑,1985,5:399-412
[16] Chow Y S. Some convergence theorems for indepent random variables. Ann. Math. Statist., 1966, 37: 1482-1493
[17]王小明. NA序列部分和的完全收敛性[J].应用数学学报, 1999,22(3): 407-412
[18]吴群英.同分布NA序列的完全收敛性[J].中国基础科学, 1999(2-4): 89-91
[19] Petrov V V.独立随机变量之和的极限定理.苏淳,黄可明译.北京:科学出版社,1987
[20]王岳宝,刘许国,苏淳.独立加权和的完全收敛性的等价条件[J].中国科学, A辑, 1998, 28(3): 213-222
[21]吴群英,王岳宝.独立阵列和的最大值完全收敛的等价条件[J].系统科学与数学,2002,22(2):192-199
[22] Zhang L. X., Wang X. Y. Convergence rates in the strong laws of asymptotically negatively associated random fields[J]. Appl. Math. J. Chinese Univ.;1999;14(4): 406–416
[23] Zhang L X. A functional central limit theorem for asymptotically negatively dependent random fields[J]. Acta Math. Hungar., 2000, 86(3): 237–259
[24] Zhang L. X. Central limit theorems for asymptotically negatively associated random fields[J]. Acta Math.Sinica, English Series.; 2000; 16(4): 691–710
[25]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(A辑), 2003, 18(3): 288-294.
[26]周慧.混合序列的矩不等式及其应用[J].浙江大学学报(理学版), 2006, 33 (6): 632-636
[27] Wang,J F, LU,F B. Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables[J]. Acta Math.Sinica, 2006, 22(3): 693-700
[28]张立新,闻继威. B值混合随机场的强大数律[J].数学年刊A辑(中文版), 2001, 22A: 2, 205-216
[29] Thrum R. A remark on almost sure convergence of weighted sums[J]. Probab. Th. Rel. Fields, 1987, 75: 425-430
[30]杨善朝.混合序列加权和的强收敛性[J].系统科学与数学, 1995, 15(3): 254-265
[31]吴本忠.ρ混合序列加权和的完全收敛性[J].高校应用数学学报, 1997, 12: 291-297
[32]蔡光辉,吴航.ρ混合序列加权和的完全收敛性[J].应用数学学报, 2005, 28 (04): 622-628 .
[33]蔡光辉,许冰.ρ混合序列加权和的完全收敛性及其应用[J].数学杂志, 2006, 26(04): 419-422
[34]吴群英.ρ?混合序列加权和的完全收敛性和强收敛性[J].应用数学,2002,15(1):1-4
[35]陈平炎.混合序列加权和的强大数定律[J].应用数学,2005,18(4):517-520
[36]朱孟虎.ρ*混合序列加权和的完全收敛性[J].纯粹数学与应用数学,2006,22(1):122-124
[37]鄢寒,吴群英,孟兵. Complete Convergence and Strong Convergence for Weighted Sums ofρ? -Mixing Random Sequences[J],广西科学,2009,16(1): 38-40,45