两类非独立随机变量的极限理论
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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。
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.
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。
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.
引文
1. B. V. Gnedenko, A. M. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Nauka, Moscow(Russion). 1949, English Edition: Reading, Mass: Addision-wesley. 1954
2. A. Gut. Precise Asympototics for Record Times and the Associated Counding Process. Stoch. Proc. Appl.. 2002, 34(10):233~239
3. E. Hopf, Ergodentheorie, Ergebnisse der Math. Springer-Verlag, Berlin. 1937: 5
4. W. Hoeffding, H. Robbins. The Central Limit Theorem for Dependent Random Variables. J Duke Math.1948, 15(1):773~780
5.陆传荣,林正炎.混合相依变量的极限理论.科学出版社. 1997
6. B. De Fintti. Funzione Caratteristica Di Unfenomeno Aleatorio. Atti Accad. NaZ. Lincei Rend. cl. Sci. Fis. Mat. Nat.. 1930, 9(4):86~133
7. D. J. Aldous. Exchangeability and Related Topics. Lecture Notes in Math, New York. Springer. 1985
8. J. Blum, H. Chernoff, M. Rosenblatt, H. Teicher. Central Limit Theorems for Interchangeable Processes. Canad. J. Math.. 1958, 9(10):122~124
9. R. L. Taylor. Laws of Large Numbers for Dependent Random Variables. Colloquia Mathematica Societatics Janos Bolyai. Limit Theorems in Probability and Statistics, Veszprem(Hungary). 1982:1023~1042
10. H. Chernoff, H. Teicher. A Central Limit Theorem for Sums of Interchangeable Random Variables. Ann.Math. Statist.. 1958, 29(2):118~130
11.王力,刘家琦.按行可交换随机向量函数加权和的收敛性.哈尔滨工业大学学报. 2001, 33(6):826~829
12. Etemadi, N. Stability of Weighted Averages of 2-Exchangeable Random Variables. Statistics & Probability letters. 2007, 77(3):389~395
13. Yutao Ma, Qiongxia Song, Liming Wu. Large Deviation Principles with Respect to the T-topology for Exchangeable Sequences: A Necessary and Sufficient Condition. Statistics & Probability letters. 2007, 77(2):239~246
14. Wu, L.. Large Deviation Principle for Exchangeable Sequences: Necessary and Sufficient Condition. J. Theoret. Probab. 2004, 17:967~978
15. A. N. Kolmogorov, U. A. Rozanov. On the Strong Mixing Conditions of a Stationary Gaussian Process. Probab. Theory Appl.. 1960, 9(2):222~227
16.邵启满.关于ρ-混合序列的完全收敛性.数学学报. 1989, 32(9):377~393
17.杨善朝.混合序列加权和的强收敛性.系统科学与数学. 1995, 15(3): 254~265
18.陆传荣.一类混合过程的弱收敛性.杭州大学学报. 1985, 12(1):7~11
19. Chen Pingyan, Gan Shixin. On Moments of the Maximum of Normed Partial Sums ofρ-mixing Random Variables. Statistics and Probability Letters. 2008, 78:1215~1221
20.邵启满.关于ρ-混合随机变量序列的不变原理的注记.数学年刊A辑. 1988, 17(9): 409~412
21. Anna Kuczmaszewska. On Complete Convergence for Arrays of Rowwise Dependent Random Variables. Statistics and Probability Letters. 2007, 77: 1050~1060
22.邵启满,陆传荣.关于相依随机变量部分和的强逼近.中国科学A辑. 1986, 5(12):1243~1253
23. Soo Hak Sung. Complete Convergence for Weighted Sums of Random Variables. Statistics and Probability Letters. 2007, 77:303~311
24.杨小云,焦志常.关于移动平均过程完全收敛性的研究.应用概率统计. 1999, 15(1):19~27
25. An Jun, Yuan Demei. Complete Convergence of Weighted Sums forρ*-mixing Sequence of Random Variables. Statistics and Probability Letters. 2008, 78:1466~1472
26.吴本忠.ρ混合序列加权和的完全收敛性.高校应用数学学报. 1997, 6(12): 291~297
27. R. C. Bradley. Equivalent Mixing Conditions for Random Fields. Technical Report No.336, Center for Stochastic Processes. 1990
28.吴群英.ρ混合序列的若干收敛性质.工程数学学报. 2001, 18(3):58~64,50
29.吴群英.ρ混合序列加权和的完全收敛性和强收敛性.应用数学. 2002, 15(1): 1~4
30.吴群英.不同分布ρ混合序列的强收敛速度.数学研究与评论. 2004, 24(1): 173~179
31. M. Klass, H. Teicher. The Central Limit Theorems for Exchangeable Random Variables without Moments. Ann. Probab.. 1987, 15:138~153
32. J. F. C. Kingman. Uses of Exchangeability. Ann. Probab.. 1978, 6:183~197
33. L. E. Baum, M. Katz. Convergence Rate in the Law of Large Numbers. Trans. Amer. Math. Soc.. 1989, 12(1):108~123
34. P. L. Hsu. H. Robbins. Complete Convergence and the Law of Large Numbers. Proc. Nat. Acad. Sci. U.S.A.. 1947, 33(2):25~31
35. K. Joag-Dev, F. Proschan. Negative Associated of Random Variables with Application. Ann. Statist.. 1983, 11(1):286~295
36. T. L. Lai, Robbins Herbert. Strong Consistency of Least Squares Estimates in Regression Models. Proc. Nat. Acad. Sci. USA..1994, 74 (6):2667~2669
37. Soo Hak Sung. A Note on the Complete Convergence for Arrays of Rowwise Independent Random Elements. Statistics and Probability Letters. 2008, 78: 1283~1289
38. Zhang Lixin. Central Limit Theorems for Asymptotically Negatively Associated Random Fields, Acta Mathematica Sinica. 2000, 16(4):691~710
39. Anna Kuczmaszewska. On Complete Convergence for Arrays of Rowwise Negatively Associated Random Variables. Statistics & Probability letters. 2009, 79(1):116~125
40.苏淳,江涛,唐启鹤.独立同分布随机变量部分和之随机和的极限定理.中国科学技术大学学报. 2001, 31(4):394~399
41.林正炎,陆传荣,苏中根.概率极限理论基础.高等教育出版社.北京, 1999:85~86
42. R. C. Bradley. On the Spectral Density and Asymptotic Normality of Weakly Dependent Random Fields. Theorem Probab..1992, (5):355~374
43. W. Bryc, W. Smolenski. Moment Condition for Almost Sure Convergence of Weakly Correlated Random Variables. Proceedings of American Math Society. 1993, 119(2):629~635
44. Tien-Chung Hu, Andrew Rosalsky, Andrei Volodin. On Convergence Properties of Sums of Dependent Random Variables under Second Moment and Covariance Rrestrictions. Statistics and Probability Letters. 2008, 78:1999~2005
45.吴群英.ρ-混合序列的不变原理.纯粹数学与应用学报. 2003, 19(1):12~15
46. R. L. Taylor, P. I. Daffer, R. F. Patterson. Limit Theorems for Sums of Exchangeable Random Variables. 1985
47. Y. S. Chow, H. Teither. Independence Interchangeability Martingales. New York: Springer-Verlag. 1978
48.吴群英.混合序列的概率极限理论.科学出版社. 2006
49. W. F. Stout. Almost Sure Convergence. New York: Academic Press. 1974
50. M. Peligard, A. Gut. Almost Sure Results for a Class of Dependent Random variables. J. Theoret Probab.. 1999, 12(1):87~104
51. S. Utev, M. Peligard. Maximal Inequalities and an Invariance Principle for a Class of Weakly Random Variabies. J. Theoret Probab.. 2003, 16(1):101~115
52. Zhang Linxin. Convergence Rates in the Strong Law of Nonstationaryρ-Mixing Random Fields. Acta Mathematica Scientia.. 2000, 20(B):303~312
53.朱孟虎,蔡光辉.混合序列加权和的完全收敛性.纯粹数学与应用数学, 2006, 22(1):122~124
2. A. Gut. Precise Asympototics for Record Times and the Associated Counding Process. Stoch. Proc. Appl.. 2002, 34(10):233~239
3. E. Hopf, Ergodentheorie, Ergebnisse der Math. Springer-Verlag, Berlin. 1937: 5
4. W. Hoeffding, H. Robbins. The Central Limit Theorem for Dependent Random Variables. J Duke Math.1948, 15(1):773~780
5.陆传荣,林正炎.混合相依变量的极限理论.科学出版社. 1997
6. B. De Fintti. Funzione Caratteristica Di Unfenomeno Aleatorio. Atti Accad. NaZ. Lincei Rend. cl. Sci. Fis. Mat. Nat.. 1930, 9(4):86~133
7. D. J. Aldous. Exchangeability and Related Topics. Lecture Notes in Math, New York. Springer. 1985
8. J. Blum, H. Chernoff, M. Rosenblatt, H. Teicher. Central Limit Theorems for Interchangeable Processes. Canad. J. Math.. 1958, 9(10):122~124
9. R. L. Taylor. Laws of Large Numbers for Dependent Random Variables. Colloquia Mathematica Societatics Janos Bolyai. Limit Theorems in Probability and Statistics, Veszprem(Hungary). 1982:1023~1042
10. H. Chernoff, H. Teicher. A Central Limit Theorem for Sums of Interchangeable Random Variables. Ann.Math. Statist.. 1958, 29(2):118~130
11.王力,刘家琦.按行可交换随机向量函数加权和的收敛性.哈尔滨工业大学学报. 2001, 33(6):826~829
12. Etemadi, N. Stability of Weighted Averages of 2-Exchangeable Random Variables. Statistics & Probability letters. 2007, 77(3):389~395
13. Yutao Ma, Qiongxia Song, Liming Wu. Large Deviation Principles with Respect to the T-topology for Exchangeable Sequences: A Necessary and Sufficient Condition. Statistics & Probability letters. 2007, 77(2):239~246
14. Wu, L.. Large Deviation Principle for Exchangeable Sequences: Necessary and Sufficient Condition. J. Theoret. Probab. 2004, 17:967~978
15. A. N. Kolmogorov, U. A. Rozanov. On the Strong Mixing Conditions of a Stationary Gaussian Process. Probab. Theory Appl.. 1960, 9(2):222~227
16.邵启满.关于ρ-混合序列的完全收敛性.数学学报. 1989, 32(9):377~393
17.杨善朝.混合序列加权和的强收敛性.系统科学与数学. 1995, 15(3): 254~265
18.陆传荣.一类混合过程的弱收敛性.杭州大学学报. 1985, 12(1):7~11
19. Chen Pingyan, Gan Shixin. On Moments of the Maximum of Normed Partial Sums ofρ-mixing Random Variables. Statistics and Probability Letters. 2008, 78:1215~1221
20.邵启满.关于ρ-混合随机变量序列的不变原理的注记.数学年刊A辑. 1988, 17(9): 409~412
21. Anna Kuczmaszewska. On Complete Convergence for Arrays of Rowwise Dependent Random Variables. Statistics and Probability Letters. 2007, 77: 1050~1060
22.邵启满,陆传荣.关于相依随机变量部分和的强逼近.中国科学A辑. 1986, 5(12):1243~1253
23. Soo Hak Sung. Complete Convergence for Weighted Sums of Random Variables. Statistics and Probability Letters. 2007, 77:303~311
24.杨小云,焦志常.关于移动平均过程完全收敛性的研究.应用概率统计. 1999, 15(1):19~27
25. An Jun, Yuan Demei. Complete Convergence of Weighted Sums forρ*-mixing Sequence of Random Variables. Statistics and Probability Letters. 2008, 78:1466~1472
26.吴本忠.ρ混合序列加权和的完全收敛性.高校应用数学学报. 1997, 6(12): 291~297
27. R. C. Bradley. Equivalent Mixing Conditions for Random Fields. Technical Report No.336, Center for Stochastic Processes. 1990
28.吴群英.ρ混合序列的若干收敛性质.工程数学学报. 2001, 18(3):58~64,50
29.吴群英.ρ混合序列加权和的完全收敛性和强收敛性.应用数学. 2002, 15(1): 1~4
30.吴群英.不同分布ρ混合序列的强收敛速度.数学研究与评论. 2004, 24(1): 173~179
31. M. Klass, H. Teicher. The Central Limit Theorems for Exchangeable Random Variables without Moments. Ann. Probab.. 1987, 15:138~153
32. J. F. C. Kingman. Uses of Exchangeability. Ann. Probab.. 1978, 6:183~197
33. L. E. Baum, M. Katz. Convergence Rate in the Law of Large Numbers. Trans. Amer. Math. Soc.. 1989, 12(1):108~123
34. P. L. Hsu. H. Robbins. Complete Convergence and the Law of Large Numbers. Proc. Nat. Acad. Sci. U.S.A.. 1947, 33(2):25~31
35. K. Joag-Dev, F. Proschan. Negative Associated of Random Variables with Application. Ann. Statist.. 1983, 11(1):286~295
36. T. L. Lai, Robbins Herbert. Strong Consistency of Least Squares Estimates in Regression Models. Proc. Nat. Acad. Sci. USA..1994, 74 (6):2667~2669
37. Soo Hak Sung. A Note on the Complete Convergence for Arrays of Rowwise Independent Random Elements. Statistics and Probability Letters. 2008, 78: 1283~1289
38. Zhang Lixin. Central Limit Theorems for Asymptotically Negatively Associated Random Fields, Acta Mathematica Sinica. 2000, 16(4):691~710
39. Anna Kuczmaszewska. On Complete Convergence for Arrays of Rowwise Negatively Associated Random Variables. Statistics & Probability letters. 2009, 79(1):116~125
40.苏淳,江涛,唐启鹤.独立同分布随机变量部分和之随机和的极限定理.中国科学技术大学学报. 2001, 31(4):394~399
41.林正炎,陆传荣,苏中根.概率极限理论基础.高等教育出版社.北京, 1999:85~86
42. R. C. Bradley. On the Spectral Density and Asymptotic Normality of Weakly Dependent Random Fields. Theorem Probab..1992, (5):355~374
43. W. Bryc, W. Smolenski. Moment Condition for Almost Sure Convergence of Weakly Correlated Random Variables. Proceedings of American Math Society. 1993, 119(2):629~635
44. Tien-Chung Hu, Andrew Rosalsky, Andrei Volodin. On Convergence Properties of Sums of Dependent Random Variables under Second Moment and Covariance Rrestrictions. Statistics and Probability Letters. 2008, 78:1999~2005
45.吴群英.ρ-混合序列的不变原理.纯粹数学与应用学报. 2003, 19(1):12~15
46. R. L. Taylor, P. I. Daffer, R. F. Patterson. Limit Theorems for Sums of Exchangeable Random Variables. 1985
47. Y. S. Chow, H. Teither. Independence Interchangeability Martingales. New York: Springer-Verlag. 1978
48.吴群英.混合序列的概率极限理论.科学出版社. 2006
49. W. F. Stout. Almost Sure Convergence. New York: Academic Press. 1974
50. M. Peligard, A. Gut. Almost Sure Results for a Class of Dependent Random variables. J. Theoret Probab.. 1999, 12(1):87~104
51. S. Utev, M. Peligard. Maximal Inequalities and an Invariance Principle for a Class of Weakly Random Variabies. J. Theoret Probab.. 2003, 16(1):101~115
52. Zhang Linxin. Convergence Rates in the Strong Law of Nonstationaryρ-Mixing Random Fields. Acta Mathematica Scientia.. 2000, 20(B):303~312
53.朱孟虎,蔡光辉.混合序列加权和的完全收敛性.纯粹数学与应用数学, 2006, 22(1):122~124