L~r（r>1）混合和ψ-混合序列的矩不等式及其应用
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摘要
近两个世纪以来,有关随机变量序列的极限定理的研究一直是概率极限理论研究的中心课题,纵览整个发展过程,给人印象深刻的是：经典问题的最终解决,不是靠工具的改进,就是靠方法的精密化,例如Cr不等式、Minkowski不等式,以及到Holder不等式等更精细的不等式.而关于随机变量序列的矩不等式及三级数定理的研究是概率极限理论研究的一个重要组成部分.近年来有许多学者在研究此类问题,并且取得了很多非常完美的结果.
ρ-混合的概念由Kolmogorov和Rozanov(1960)引入,φ-混合的概念由Dobrushin(1956)首先在研究马氏链过程中引入,并且取得许多重要结果.ρ-混合是在1990年由Bradley引入,有关φ-混合的知识我们可在吴群英2004年的文献中找到.两两NQD序列首先是由Lehmann(1966)提出的,吴群英在文献[8]对其做了较细致的研究.近些年来,已有许多作者研究了Lr-混合鞅的性质,可参见文献Hall和Heyde(1980)、Hansen(1991)和潘等(2003).如果{Xi,i≥1}是一列鞅差序列,并且在Lp空间上有界,Lesigne和Volny在2001年得到对p≥2的估计：μ(Sn>n)≤Cn-p/2.Li在2003年将此估计推广到1n)≤Cn1-p.Hu和Wang(2008)讨论了φ-混合序列、NA序列和线性过程的大偏差估计.
本文主要总结了独立随机变量序列、ρ-混合序列、两两NQD序列、NA序列的三级数定理,并加以证明.通过证明过程可以看出吴群英在文献[8]中关于两两NQD序列推广的Kolmogorov型不等式存在系数错误.本文给予纠正.同时,我们将给出Lr(r>1)混合鞅的矩不等式,其中混合系数将具体给出,并得到了Lr(r>1)混合鞅的大偏差结果.这一结果与Lesigne和Volny(2001)、Li(2003)等人给出的关于鞅差序列的上界相同,在主要数量级上已达最优.顺便我们将其应用到线性过程中,得到了线性过程的大偏差结果.
最后,本文借鉴已有的三级数定理的研究方法给出了Ψ-混合序列矩不等式并加以应用得到了Ψ-混合序列的三级数定理.进一步完善了有关混合序列的三级数定理.
Since the last two centuries, the limit theorem is the central subject of the probability limit theory. Through the whole evolution, what made people deeply impressed is that the final settlement of the classical problems, which mainly depends on the improvement of the tools and the precision of the method, such as, Cr inequality, Minkowski's inequality, Holder's inequality, etc. The study of the moment inequality and the three theorem for random variables sequence is an important part of the probability limit theory, which has already been studied by many scholars. Particularly, a lot of perfect results have been obtained.
p-mixing random variables andφ-mixing random variables were introduced by Kolmogorov and Rozanov(1960) and Dobrushin(1956) for non-stationary Markov chain, respectively. A number of writers have studied them and established a series of useful results,ρ-mixing random variables were introduced by Bradley in 1990 andφ-mixing random variables can be found in Wu s artial which was published in 2004. The pairwise NQD sequence was introduced by Lehmann(1966) and Wu Qunying studied it deeply in [8]. In the last few years, many writers have studied qualities of Lr-mixingale sequence, for example, Hall and Heyde(1980), Hansen(1991) and Pan, et al(2003). Let{Xi,i≥1} is a bounded difference martingale sequence in Lp space. Lesigne and Volny(2001) obtained estimator:μ(Sn> n)≤Cn-p/2 for p≥2. Li(2003) generalized the result, and got that/μ(Sn> n)≤Cn1-p for 1 In this pape, we summarize the three series theorems for independent random variable sequence,ρ-mixing sequence, the pair wise NQD sequence and NA sequence, and give some proofs. Though the proof of NQD sequence, we correct the coefficient fact of Kolmogorov inequality for NQD sequence in Wu. At the same time, we obtain the moment inequality for Lr(r>1)mixingale, which has the concrete coefficients, and get the large deviations fo Lr(r>1)mixingale. These results have the same order optimal upper bounds for martingale difference sequence in Lesigne and Volny(2001) and Li(2003). In addition, we give its application to the linear process and obtain a large deviation for it.
At last, inspired by the currents method of three series theorem, moment inequality forψ/-mixing sequence is obtained. By using this inequality, we get the three series theorem forψ/-mixing sequence.
ρ-混合的概念由Kolmogorov和Rozanov(1960)引入,φ-混合的概念由Dobrushin(1956)首先在研究马氏链过程中引入,并且取得许多重要结果.ρ-混合是在1990年由Bradley引入,有关φ-混合的知识我们可在吴群英2004年的文献中找到.两两NQD序列首先是由Lehmann(1966)提出的,吴群英在文献[8]对其做了较细致的研究.近些年来,已有许多作者研究了Lr-混合鞅的性质,可参见文献Hall和Heyde(1980)、Hansen(1991)和潘等(2003).如果{Xi,i≥1}是一列鞅差序列,并且在Lp空间上有界,Lesigne和Volny在2001年得到对p≥2的估计：μ(Sn>n)≤Cn-p/2.Li在2003年将此估计推广到1
本文主要总结了独立随机变量序列、ρ-混合序列、两两NQD序列、NA序列的三级数定理,并加以证明.通过证明过程可以看出吴群英在文献[8]中关于两两NQD序列推广的Kolmogorov型不等式存在系数错误.本文给予纠正.同时,我们将给出Lr(r>1)混合鞅的矩不等式,其中混合系数将具体给出,并得到了Lr(r>1)混合鞅的大偏差结果.这一结果与Lesigne和Volny(2001)、Li(2003)等人给出的关于鞅差序列的上界相同,在主要数量级上已达最优.顺便我们将其应用到线性过程中,得到了线性过程的大偏差结果.
最后,本文借鉴已有的三级数定理的研究方法给出了Ψ-混合序列矩不等式并加以应用得到了Ψ-混合序列的三级数定理.进一步完善了有关混合序列的三级数定理.
Since the last two centuries, the limit theorem is the central subject of the probability limit theory. Through the whole evolution, what made people deeply impressed is that the final settlement of the classical problems, which mainly depends on the improvement of the tools and the precision of the method, such as, Cr inequality, Minkowski's inequality, Holder's inequality, etc. The study of the moment inequality and the three theorem for random variables sequence is an important part of the probability limit theory, which has already been studied by many scholars. Particularly, a lot of perfect results have been obtained.
p-mixing random variables andφ-mixing random variables were introduced by Kolmogorov and Rozanov(1960) and Dobrushin(1956) for non-stationary Markov chain, respectively. A number of writers have studied them and established a series of useful results,ρ-mixing random variables were introduced by Bradley in 1990 andφ-mixing random variables can be found in Wu s artial which was published in 2004. The pairwise NQD sequence was introduced by Lehmann(1966) and Wu Qunying studied it deeply in [8]. In the last few years, many writers have studied qualities of Lr-mixingale sequence, for example, Hall and Heyde(1980), Hansen(1991) and Pan, et al(2003). Let{Xi,i≥1} is a bounded difference martingale sequence in Lp space. Lesigne and Volny(2001) obtained estimator:μ(Sn> n)≤Cn-p/2 for p≥2. Li(2003) generalized the result, and got that/μ(Sn> n)≤Cn1-p for 1
At last, inspired by the currents method of three series theorem, moment inequality forψ/-mixing sequence is obtained. By using this inequality, we get the three series theorem forψ/-mixing sequence.
引文
[1]Hall P. Heyde C. C.. Martingale Limit Theory and Its Application[M]. New York, Academic Press,1980:17-29.
[2]Hansen B. E.. Strong laws for dependent heterogeneous processes[J]. Econometric Theory.1991,7:213-221.
[3]潘光明,胡舒合,方利宝,程正东.半参数回归模型估计的平均相合性[J].数学物理学报.2003,23A(5)：598-606.
[4]Lesigne E. Volny D.. Large deviations for martingales[J]. Stochastic. Process. Appl.2001,96:143-159.
[5]Li Y. L.. A martingale inequality and large deviations[J]. Statistics Probability Letters,2003,62:317-321.
[6]Hu S. H. Wang X. J.. Large deviations for some dependent sequences[J]. Acta Mathematica Scientica.2008,28(B):295-300.
[7]Tran L. T.. Roussas G. G. Yakowitz, S. Troung, V. Fixed-desigh regression for linear time series[J]. Ann. Statist.,1996,24:975-991.
[8]吴群英.混合序列的概率极限理论[M].北京,科学出社,2006：5-7,170-225.
[9]林正炎,白志东.概率不等式[M].北京,科学出版社,2006：47,79-81.
[10]Stout W. F.. Almost Sure Convergence[M]. New York, Academic Press, 1974:24
[11]Lehmann E. L.. Some concepts of dependence[J]. Ann. Statistics.,1966, 43:1137-1153.
[12]李贤平.概率论基础[M].北京,高等教育出版社,1997：294-296.
[13]Loeve M.. Probability Theory I[M]. New York, Spring-Verlag Inc.,1977.
[14]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京,高等教育出版社,1999：87-95.
[15]吴群英.ρ混合序列加权和的完全收敛性质和强收敛性[J].应用数学,2002,15(1)：1-4.
[16]吴群英.两两NQD列的收敛性质[J].数学学报,2002,45(3)：617-624.
[17]严士健,王隽骧,刘秀芳.概率论基础[M].北京,科学出版社,1982： 446-447.
[18]吴群英.ρ混合序列的若干收敛性质[J].工程数学学报,2001,18(3)：58-64.
[19]Matula P.. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics Probability Letters,1992,15(3): 209-213.
[20]胡舒合,潘光明,高启兵.误差为线性过程时回归模型的估计问题[J].高校应用数学学报A辑,2003,18(1)：81-90.
[21]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,1997.6-10.
[22]Shao Q. M.. Almost sure invariance principles for mixing sequences of random variables[J]. Stoc. Proc. Appl.,1993,48:319-334.
[23]Kolmogorov A. N. and Rozanov U. A.. On the strong mixing conditions of a stationary Gaussian process[J]. Probab. Theory Appl.,1960,2: 222-227.
[24]Dobrushin R. L.. The central limit theorem for non-stationary Markov chain[J]. Probab. Theory Appl.,1956,1:72-88.
[25]邵启满.关于ρ-混合序列的完全收敛性[J].数学学报,1989,32：377-393.
[26]邵启满.一矩不等式及其应用[J].数学学报,1998,31：736-747.
[27]邵启满,陆传荣.关于相依随机变量部分和的强逼近[J].中国科学,A辑,1986,12：1243-1253.
[28]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.
[29]杨善朝.φ-混合序列加权和收敛性质[J].应用概率统计,1995,11(3)：273-281.
[30]吴群英.混合序列的强收敛性[J].数学研究,1999,32(1)：92-97.
[31]Bradley R. C.. Equivalent mixing conditions for random fields. Technical Report No.336, Center for Stochastic Processes, Univ. of North Carolina, Chapel Hill,1991.
[32]吴群英,林亮.φ混合序列的完全收敛性和强收敛性[J].工程数学学报,2004,21(1)：75-80.
[33]Utev S., Peligrad M..Maximal inequalities and an invariance principle for a class of weakly dependent random variables [J]. J. Theoret. Probab., 2003,16(1):101-115.
[34]Joag-Dev K., Proschan F.. Negative association of random variables with applications[J]. Ann. Statist.,1983,11(1):286-295
[2]Hansen B. E.. Strong laws for dependent heterogeneous processes[J]. Econometric Theory.1991,7:213-221.
[3]潘光明,胡舒合,方利宝,程正东.半参数回归模型估计的平均相合性[J].数学物理学报.2003,23A(5)：598-606.
[4]Lesigne E. Volny D.. Large deviations for martingales[J]. Stochastic. Process. Appl.2001,96:143-159.
[5]Li Y. L.. A martingale inequality and large deviations[J]. Statistics Probability Letters,2003,62:317-321.
[6]Hu S. H. Wang X. J.. Large deviations for some dependent sequences[J]. Acta Mathematica Scientica.2008,28(B):295-300.
[7]Tran L. T.. Roussas G. G. Yakowitz, S. Troung, V. Fixed-desigh regression for linear time series[J]. Ann. Statist.,1996,24:975-991.
[8]吴群英.混合序列的概率极限理论[M].北京,科学出社,2006：5-7,170-225.
[9]林正炎,白志东.概率不等式[M].北京,科学出版社,2006：47,79-81.
[10]Stout W. F.. Almost Sure Convergence[M]. New York, Academic Press, 1974:24
[11]Lehmann E. L.. Some concepts of dependence[J]. Ann. Statistics.,1966, 43:1137-1153.
[12]李贤平.概率论基础[M].北京,高等教育出版社,1997：294-296.
[13]Loeve M.. Probability Theory I[M]. New York, Spring-Verlag Inc.,1977.
[14]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京,高等教育出版社,1999：87-95.
[15]吴群英.ρ混合序列加权和的完全收敛性质和强收敛性[J].应用数学,2002,15(1)：1-4.
[16]吴群英.两两NQD列的收敛性质[J].数学学报,2002,45(3)：617-624.
[17]严士健,王隽骧,刘秀芳.概率论基础[M].北京,科学出版社,1982： 446-447.
[18]吴群英.ρ混合序列的若干收敛性质[J].工程数学学报,2001,18(3)：58-64.
[19]Matula P.. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics Probability Letters,1992,15(3): 209-213.
[20]胡舒合,潘光明,高启兵.误差为线性过程时回归模型的估计问题[J].高校应用数学学报A辑,2003,18(1)：81-90.
[21]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,1997.6-10.
[22]Shao Q. M.. Almost sure invariance principles for mixing sequences of random variables[J]. Stoc. Proc. Appl.,1993,48:319-334.
[23]Kolmogorov A. N. and Rozanov U. A.. On the strong mixing conditions of a stationary Gaussian process[J]. Probab. Theory Appl.,1960,2: 222-227.
[24]Dobrushin R. L.. The central limit theorem for non-stationary Markov chain[J]. Probab. Theory Appl.,1956,1:72-88.
[25]邵启满.关于ρ-混合序列的完全收敛性[J].数学学报,1989,32：377-393.
[26]邵启满.一矩不等式及其应用[J].数学学报,1998,31：736-747.
[27]邵启满,陆传荣.关于相依随机变量部分和的强逼近[J].中国科学,A辑,1986,12：1243-1253.
[28]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.
[29]杨善朝.φ-混合序列加权和收敛性质[J].应用概率统计,1995,11(3)：273-281.
[30]吴群英.混合序列的强收敛性[J].数学研究,1999,32(1)：92-97.
[31]Bradley R. C.. Equivalent mixing conditions for random fields. Technical Report No.336, Center for Stochastic Processes, Univ. of North Carolina, Chapel Hill,1991.
[32]吴群英,林亮.φ混合序列的完全收敛性和强收敛性[J].工程数学学报,2004,21(1)：75-80.
[33]Utev S., Peligrad M..Maximal inequalities and an invariance principle for a class of weakly dependent random variables [J]. J. Theoret. Probab., 2003,16(1):101-115.
[34]Joag-Dev K., Proschan F.. Negative association of random variables with applications[J]. Ann. Statist.,1983,11(1):286-295