随机变量序列部分和乘积的几乎处处中心极限定理
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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。
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.
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。
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.
引文
1 G. Brosamler. An Almost Everywhere Central Limit Theorem.Math Pro Cambridge Philes Soc.1988, 104(3):561~574
2 P. Schatte. On Strong Versions of The Central Limit Theorem. Math. Nachr. 1988, 137(4):249 ~256
3 I. Berkes, H. Dehling, T. F. Móri. Counter Examples Related to The a.s. Central Limit Theorem. Studia Sci.Math.Hungar. 1991, 26(2):153~164
4 M. Lacey, W. Philipp. A Note on The Almost Sure Central Limit Theorem. Statist.Probab.Lett. 1990, 9(3):201~205
5 I. Berkes, H. Dehling. Some Limit Theorems in Log Density. Ann. Probab. 1993, 21(3):1640 ~1670
6 I. Berkes, E. Csaki. A Universal Result in Almost Sure Central Limit Theory. Stochastic Process. Appl. 2001, 94(6):105~134
7 M. Peligrad, Q. M. Shao. A Note on The Almost Sure Central Limit Theorem for Weakly Dependent Random Variables. Statist.Probab.Lett. 1995, 22(2):131~136
8 P. Matula. On The Almost Sure Central Limit Theorem for Associated Random Variable. Probab.Math.Statist. 1998, 18(2):411~416
9 I. Berkes. Results and Problems Related to The Pointwise Central Limit Theorem. Asymptotic Methods in Probability and Statistics, Szyszkowicz. B.(Ed), Elsevier, Amsterdam. 1998:59~96
10 B. C. Arnold, J. A. Villasenor. The Asymptotic Distribution of Sums of Records. Extremes. 1998, 1(3):351~363
11 G. Rempala, J. Wesolowski. Asymptotics for Products of Sums and U -Statistics. Elect Comm in probab. 2002, 7(2):47~54
12 Khurelbaatar, Grzegorz. A Note on The Almost Sure Central Limit Theorem for The Product of Partial Sums.//IMA Preprint Series 1968.Minnesota: University of Minnesota, 2004
13胡星,徐杉. ? -混合序列部分和乘积的几乎处处中心极限定理.浙江大学学报(理学版). 2007, 34(5):505~508
14金敬森.强混合序列部分和乘积的几乎处处中心极限定理.浙江大学学报(理学版). 2007, 34(1):24~27
15陈守全,林正炎.随机变量函数序列的几乎处处中心极限定理.数学物理学报. 2008,28A(4):747~756
16 Csaki, Khurelbaatar. Almost Sure Limit Theorems for The Maximum of Stationary Gaussian Sequences. Statist.Probab.Lett. 2002, 58(2):195~203
17 Fahrner, Stadtmuller. On Almost Sure Max-Limit Theorems. Stat. Prob. Lett. 1998, 37(3): 229~236
18 Tong Bin, Peng Zuo-xiang, Zhao Sheng-li. Almost Sure Versions of Central Limit Theorems for Order Statistics.西南大学学报(自然科学版). 2008, 3(30):20~24
19王芳,程士宏. U -统计量的几乎处处中性极限定理.数学年刊. 2003, 24A(6):735~742
20董志山,杨小云. NA及LNQD随机变量列的几乎处处中心极限定理.数学学报. 2004, 47(3):593~600
21 G. Hurelbaatar. Almost Sure Limit Theorems for Dependent Random Variables. Studia Sci. Math. Hungar. 1997, 33(1):167~175
22 G. Khurelbaatar, G. Rempala. A Note on Almost Sure Limit Theorem for The Product of Partial Sums. Applied Mathematics Letters. 2006, 19(7):191~196
23 G. Khurelbaatar. Almost Sure Central Limit Theorem. University of Cincinnati Ph.D. Dissertation. 2001
24 E. L. Lehmann. Some Concepts of Dependence. Ann. Math. Statist. 1966, 37(5):1137~1153
25周慧.ρ? -混合序列几乎处处中心极限定理的注记.浙江大学学报(理学版).2005, 32(5) :503~505
26 Doukhan, Louhichi. A New Dependence Condition and Applications to Moment Inequalities. Stochastic processes and their applications. 1999, (84):313~342
27 Marcin Dudzinski. A Note on Almost Sure Central Limit Theorem for Some Dependent Random Variables. Statistics & Probability Letters. 2003, (61):31~40
28 Emmanuel Lesigne. Almost Sure Central Limit Theorem for Strictly Stationary Processes. AMS. 2000, (128):1751~1759
29 Cheng S., Peng L. and Qi Y. Almost Sure Convergence in Extreme Value Theory. Math. Nachr. 1998, (190):43~50
30陈志成,彭作祥.平稳高斯向量序列最大值的几乎处处中心极限定理.西南大学学报(自然科学版). 2007, 3(29):23~27
31李云霞.由渐近线性坐标负相依产生的平稳线性过程的泛函中心极限定理.浙江大学学报(理学版). 2003, 30(5):495~498
32 Zhang L. X. Central Limit Theorems for Asymptotically Negatively Associated Random Fields. Acta Math Sinica. 2000, 16(4):691~710
33 Zhang L. X. A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Variables. Acta. Math Hunger. 2000, 86(3):237~259
34 Qi Y. Limit Distributions for Pruducts of Sums. Statist.Probab.Lett. 2003, 62(1):92~100
35 Qi Y. A Note on Asymptotic Distributions for Products of Sums. Statist Probab Lett. 2004, 68(4):407~413
36胡星. ? -混合序列部分和乘积的渐近正态性.浙江大学学报(理学版). 2006, 33(3): 255~259
37 P. Magda, U. Sergey. Central Limit Theorem for Linear Processes. Ann. Probab. 1997, 25(1): 454
38 J. Dedecker, P. Doukhan. A New Covariance Inequality and Applications. Stoch. Proc. Appl. 2003, (106):63~80
39 Stadtmuller. Almost Sure Versions of Distributional Limit Theorems for Certain Order Statistics. Statist.Probab.Lett. 2002, 58(8):413~426
40 M. R. Leadbetter, G. Lindgren, H. Rootzen. Extremes and Related Properties of Random Sequences and Processes. Berlin:Springer, 1983
41 Chen S. Q., Lin Z. Y. Almost Sure Max-Limit for Nonstationary Gaussian Sequence. Statist Probab.Lett. 2006, 76(11):1175~1184
42陆传荣,林正炎.混合相依随机变量的极限理论.科学出版社, 1997
43吴群英.混合序列的概率极限理论.科学出版社, 2006
44王芳.均匀经验过程的几乎处处中心极限定理.首都师范大学学报(自然科学版). 2005, 26(2):9~11
45林正炎,苏中根,张立新译.当前概率学科中的研究机遇.数学进展. 2004, 33(2):129~ 140
2 P. Schatte. On Strong Versions of The Central Limit Theorem. Math. Nachr. 1988, 137(4):249 ~256
3 I. Berkes, H. Dehling, T. F. Móri. Counter Examples Related to The a.s. Central Limit Theorem. Studia Sci.Math.Hungar. 1991, 26(2):153~164
4 M. Lacey, W. Philipp. A Note on The Almost Sure Central Limit Theorem. Statist.Probab.Lett. 1990, 9(3):201~205
5 I. Berkes, H. Dehling. Some Limit Theorems in Log Density. Ann. Probab. 1993, 21(3):1640 ~1670
6 I. Berkes, E. Csaki. A Universal Result in Almost Sure Central Limit Theory. Stochastic Process. Appl. 2001, 94(6):105~134
7 M. Peligrad, Q. M. Shao. A Note on The Almost Sure Central Limit Theorem for Weakly Dependent Random Variables. Statist.Probab.Lett. 1995, 22(2):131~136
8 P. Matula. On The Almost Sure Central Limit Theorem for Associated Random Variable. Probab.Math.Statist. 1998, 18(2):411~416
9 I. Berkes. Results and Problems Related to The Pointwise Central Limit Theorem. Asymptotic Methods in Probability and Statistics, Szyszkowicz. B.(Ed), Elsevier, Amsterdam. 1998:59~96
10 B. C. Arnold, J. A. Villasenor. The Asymptotic Distribution of Sums of Records. Extremes. 1998, 1(3):351~363
11 G. Rempala, J. Wesolowski. Asymptotics for Products of Sums and U -Statistics. Elect Comm in probab. 2002, 7(2):47~54
12 Khurelbaatar, Grzegorz. A Note on The Almost Sure Central Limit Theorem for The Product of Partial Sums.//IMA Preprint Series 1968.Minnesota: University of Minnesota, 2004
13胡星,徐杉. ? -混合序列部分和乘积的几乎处处中心极限定理.浙江大学学报(理学版). 2007, 34(5):505~508
14金敬森.强混合序列部分和乘积的几乎处处中心极限定理.浙江大学学报(理学版). 2007, 34(1):24~27
15陈守全,林正炎.随机变量函数序列的几乎处处中心极限定理.数学物理学报. 2008,28A(4):747~756
16 Csaki, Khurelbaatar. Almost Sure Limit Theorems for The Maximum of Stationary Gaussian Sequences. Statist.Probab.Lett. 2002, 58(2):195~203
17 Fahrner, Stadtmuller. On Almost Sure Max-Limit Theorems. Stat. Prob. Lett. 1998, 37(3): 229~236
18 Tong Bin, Peng Zuo-xiang, Zhao Sheng-li. Almost Sure Versions of Central Limit Theorems for Order Statistics.西南大学学报(自然科学版). 2008, 3(30):20~24
19王芳,程士宏. U -统计量的几乎处处中性极限定理.数学年刊. 2003, 24A(6):735~742
20董志山,杨小云. NA及LNQD随机变量列的几乎处处中心极限定理.数学学报. 2004, 47(3):593~600
21 G. Hurelbaatar. Almost Sure Limit Theorems for Dependent Random Variables. Studia Sci. Math. Hungar. 1997, 33(1):167~175
22 G. Khurelbaatar, G. Rempala. A Note on Almost Sure Limit Theorem for The Product of Partial Sums. Applied Mathematics Letters. 2006, 19(7):191~196
23 G. Khurelbaatar. Almost Sure Central Limit Theorem. University of Cincinnati Ph.D. Dissertation. 2001
24 E. L. Lehmann. Some Concepts of Dependence. Ann. Math. Statist. 1966, 37(5):1137~1153
25周慧.ρ? -混合序列几乎处处中心极限定理的注记.浙江大学学报(理学版).2005, 32(5) :503~505
26 Doukhan, Louhichi. A New Dependence Condition and Applications to Moment Inequalities. Stochastic processes and their applications. 1999, (84):313~342
27 Marcin Dudzinski. A Note on Almost Sure Central Limit Theorem for Some Dependent Random Variables. Statistics & Probability Letters. 2003, (61):31~40
28 Emmanuel Lesigne. Almost Sure Central Limit Theorem for Strictly Stationary Processes. AMS. 2000, (128):1751~1759
29 Cheng S., Peng L. and Qi Y. Almost Sure Convergence in Extreme Value Theory. Math. Nachr. 1998, (190):43~50
30陈志成,彭作祥.平稳高斯向量序列最大值的几乎处处中心极限定理.西南大学学报(自然科学版). 2007, 3(29):23~27
31李云霞.由渐近线性坐标负相依产生的平稳线性过程的泛函中心极限定理.浙江大学学报(理学版). 2003, 30(5):495~498
32 Zhang L. X. Central Limit Theorems for Asymptotically Negatively Associated Random Fields. Acta Math Sinica. 2000, 16(4):691~710
33 Zhang L. X. A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Variables. Acta. Math Hunger. 2000, 86(3):237~259
34 Qi Y. Limit Distributions for Pruducts of Sums. Statist.Probab.Lett. 2003, 62(1):92~100
35 Qi Y. A Note on Asymptotic Distributions for Products of Sums. Statist Probab Lett. 2004, 68(4):407~413
36胡星. ? -混合序列部分和乘积的渐近正态性.浙江大学学报(理学版). 2006, 33(3): 255~259
37 P. Magda, U. Sergey. Central Limit Theorem for Linear Processes. Ann. Probab. 1997, 25(1): 454
38 J. Dedecker, P. Doukhan. A New Covariance Inequality and Applications. Stoch. Proc. Appl. 2003, (106):63~80
39 Stadtmuller. Almost Sure Versions of Distributional Limit Theorems for Certain Order Statistics. Statist.Probab.Lett. 2002, 58(8):413~426
40 M. R. Leadbetter, G. Lindgren, H. Rootzen. Extremes and Related Properties of Random Sequences and Processes. Berlin:Springer, 1983
41 Chen S. Q., Lin Z. Y. Almost Sure Max-Limit for Nonstationary Gaussian Sequence. Statist Probab.Lett. 2006, 76(11):1175~1184
42陆传荣,林正炎.混合相依随机变量的极限理论.科学出版社, 1997
43吴群英.混合序列的概率极限理论.科学出版社, 2006
44王芳.均匀经验过程的几乎处处中心极限定理.首都师范大学学报(自然科学版). 2005, 26(2):9~11
45林正炎,苏中根,张立新译.当前概率学科中的研究机遇.数学进展. 2004, 33(2):129~ 140