几类混合随机变量序列部分和的方差估计
摘要
笔者通过广泛阅读国内外关于混合序列部分和的方差估计问题的研究文献,对ρ混合序列、ρ*混合序列和α混合序列部分和的方差估计的相合性和渐进正态性的重要研究成果加以思考和比较形成此文。第一章对该问题的发展现状进行了描述,第二章主要针对平稳ρ混合序列的情况,总结研究了Bn ,p及B?n ,p是σ的相合估计,并给出了它们的渐近正态性;第三章讨论了平稳ρ*混合序列部分和估计的大样本性质,即ρ*混合随机变量序列部分和的方差估计的相合性和渐进正态性;第四章主要针对平稳α混合(强混合)序列的情况,总结研究了强混合的渐近正态性和相合性。
The random variables or stochastic processes that come from practical problems are usually not independent ,we remark that there has been a great amount of work on the properties of mixing random variables ,and some significant results have been reached through deep research by a lot of scholars. The mixing dependence means that the considered random variables are asymptotic independent, the dependence of random variables as a concept is developed in some branches of Probability theory and Mathematical Statistics,such as Markov chains,random field theory and time series analysis,etc.There is a lot of concept of mixing to measure the dependence of non-independent random variables,for example,ρ-mixing ,ρ*-mixing ,α-mixing(strong mixing),β-mixing,φ-mixing, etc.
This thesis considers three kinds of mixing dependent random variables,ρ-mixing ,ρ*-mixing ,α-mixing random variables,since this three kinds of mixing dependent random variables has applications in the theory of statistics,and the properties of that have drawn many scholars’attentions,Hence one can see that, the study on the partical sums of dependent random variables has momentous significance.Based on reading and understanding the literacture on consistency and the asymptotic normalities of the estimation of the variance for the partial sums of some mixing random variable sequences ,we synthesize and think the research results of the question.
The limit theory for the partial sums of some mixing random variable sequences has an important status in Probability and Mathematical Statistics,it also has certain research value and practical significance .For example ,for a stationary sequence be its partical sum process, let { X n(t );n∈N} be the influence of the nth investor on the return rate of a stock ,then the process describes the flucturation of the rate of return of a stock. So ,many scholars devote themselves to the research of this properties.Such as Xiao and Zheng gave the convergence of the partial sums of random processes;Lin and Lu proved that limit theory for mixing dependent random variables;Pligrad.M. gave on the central limit theorem forρ-mixing sequences of random variables;Dong and Yang proved that an almost sure central limit theorem for NA and LNQD random variables;Yang gave the strong law of large numbers forα-mixing sequences of random variables,etc.
It is well known ,in the research of limit theorems of Probability theory for a sequence of random variables,let { X n, n≥1} be a sequence of i.i.d. random variables ,σ2 is the variance of X n,there has already a lot of methods used in the estimation of the variance so far.The most usual estimation is the variance of sample ,and X = ( 1n )∑i=n1 Xi is satisfied;let { X n, n≥1} be a mixing dependent random variables, the variance of sample is not the consistent estimation ofσ2,Under some mild conditions, we have i.e.,if l is large enough,Var( Sll ) is the approximateσ2, is the estimator of Var( Sll ),let {l n , n≥1} be a sequence of positive integers ,with Bn ,2 is the variance of sample ??? s j(l ) l; j = 0,1, ???, n ? l???.Paligrad and Shao defined two sample estimators Bn ,p and B?n ,pofσ(for definition see chapterⅡ),and studied their asymptotic properties forρ-mixing random variables;In chapterⅢwe study the large sample properties of the estimators of the partical sum of the stationaryρ*-mixing sequence and get the consistency as well as the asymptotic normality;The chapterⅣanalyzes and surveys the results about the consistency and the asymptotic normality of the estimators of the variance forα-mixing sequence of random variables.
The random variables or stochastic processes that come from practical problems are usually not independent ,we remark that there has been a great amount of work on the properties of mixing random variables ,and some significant results have been reached through deep research by a lot of scholars. The mixing dependence means that the considered random variables are asymptotic independent, the dependence of random variables as a concept is developed in some branches of Probability theory and Mathematical Statistics,such as Markov chains,random field theory and time series analysis,etc.There is a lot of concept of mixing to measure the dependence of non-independent random variables,for example,ρ-mixing ,ρ*-mixing ,α-mixing(strong mixing),β-mixing,φ-mixing, etc.
This thesis considers three kinds of mixing dependent random variables,ρ-mixing ,ρ*-mixing ,α-mixing random variables,since this three kinds of mixing dependent random variables has applications in the theory of statistics,and the properties of that have drawn many scholars’attentions,Hence one can see that, the study on the partical sums of dependent random variables has momentous significance.Based on reading and understanding the literacture on consistency and the asymptotic normalities of the estimation of the variance for the partial sums of some mixing random variable sequences ,we synthesize and think the research results of the question.
The limit theory for the partial sums of some mixing random variable sequences has an important status in Probability and Mathematical Statistics,it also has certain research value and practical significance .For example ,for a stationary sequence be its partical sum process, let { X n(t );n∈N} be the influence of the nth investor on the return rate of a stock ,then the process describes the flucturation of the rate of return of a stock. So ,many scholars devote themselves to the research of this properties.Such as Xiao and Zheng gave the convergence of the partial sums of random processes;Lin and Lu proved that limit theory for mixing dependent random variables;Pligrad.M. gave on the central limit theorem forρ-mixing sequences of random variables;Dong and Yang proved that an almost sure central limit theorem for NA and LNQD random variables;Yang gave the strong law of large numbers forα-mixing sequences of random variables,etc.
It is well known ,in the research of limit theorems of Probability theory for a sequence of random variables,let { X n, n≥1} be a sequence of i.i.d. random variables ,σ2 is the variance of X n,there has already a lot of methods used in the estimation of the variance so far.The most usual estimation is the variance of sample ,and X = ( 1n )∑i=n1 Xi is satisfied;let { X n, n≥1} be a mixing dependent random variables, the variance of sample is not the consistent estimation ofσ2,Under some mild conditions, we have i.e.,if l is large enough,Var( Sll ) is the approximateσ2, is the estimator of Var( Sll ),let {l n , n≥1} be a sequence of positive integers ,with Bn ,2 is the variance of sample ??? s j(l ) l; j = 0,1, ???, n ? l???.Paligrad and Shao defined two sample estimators Bn ,p and B?n ,pofσ(for definition see chapterⅡ),and studied their asymptotic properties forρ-mixing random variables;In chapterⅢwe study the large sample properties of the estimators of the partical sum of the stationaryρ*-mixing sequence and get the consistency as well as the asymptotic normality;The chapterⅣanalyzes and surveys the results about the consistency and the asymptotic normality of the estimators of the variance forα-mixing sequence of random variables.
引文
[1] 陆传荣.林正炎,混合相依变量的极限定理.北京:科学出版社,1997
[2] Kolmogorov, A.N.,Rozanov,U.A.,On the strong mixing conditions of a stationary Gaussian process ,Probab.Theory Appl.,1960(2),222-227. (In Russian)
[3] Peligrad M. Invariance principles for mixing sequenees of random variables,Ann.Prob.,1982 , 10 , 968-981
[4] PeligradM.On the central limit theorem for mixing sequences of random variables· Ann.Prob.,1987,15 , 1387-1394
[5] Lai T L.,Convergence rates and r- quick versions of the strong law for Stationary mixing sequences, Ann.Prob.,1977,5 , 693-706
[6] 邵启满,关于ρ -混合不变原理的注记.数学年刊,1988,9,409-412
[7] Peligrad M.,Convergence rates of the strong law of stationary mixing sequences, Z.Wahr. Verw. Gebiete,1985,70 , 307 一 314
[8]苏淳,关于一类强平稳 mixing 序列完全收敛性条件的探讨,应用概率统计,1988,4, 148 一 156
[9] 孔繁超.张明俊,关于ρ -混合序列的完全收敛性的注记.科学通报,1994,39(9),778-781
[10] 孔繁超, ρ -混合序列的完全收敛性,数学年刊(A)辑,1993,l(2),159-162
[11] 王晓明,非平稳- ρ 混合序列的完全收敛性,数学学报,1996,39(4),463-476
[12] Bradley,R.C.,Equivalent mixing conditions for random fields, Technical Report No .336: Center for Stochastic Processes ,Univ.of North Csrolina,Chapel Hill, 1990.
[13] Bradley,R.C., On the spectral density and asymptotic normality of weakly dependent random fields,J.Theor.Probab., 1992(5), 355-373.
[14] Peligard,M.,On the asymptotic normality of sequences of weak dependent random varibaels,J.Theoret.Probab.,1996(9),703-715.
[15] Peligrad,M.,Maximum of partial sums and an invariance principle for a class of weak dependent random variables,Proc.AMS.,1998,126(4), 181-1189.
[16] Utev.S.,Peligard,M.,Maximal inequalities an invariance principle for a class of weakly dependent random fields,J.Theor.Probab.,2003,16(1),101-115.
[17] Bradley,R.C.,Equivalent mixing conditions for random fields, Technical Report No .336: Center for Stochastic Processes ,Univ.of North Csrolina,Chapel Hill, 1990.
[18] Miller , C.Three theorems on ρ *mixing random fields.J.of Theoert.Probab.,1994,7(4):867-882.
[19]Miller , C.A CTL for periodgrams of a ρ *mixingrandom fields.Stoch.Proc.Appl.,1995,60: 313-330.
[20] Bryc,W.;Smolenski,W.Moment conditions for almost sure convergence of weakly cor-related random varibales.Porc.Amer.Math.Soc.,1993,119(2):629-635.
[21] 杨善朝.一类随机变量部分和的矩不等式及其应用.科学通报,1998,43(17):1836-182.7
[22] 张立新,B值强混合随机场的进一步矩不等式及强大数律,应用数学学报,2000,23(4), 518-525.
[23] 张立新,闻继威.弱负相关随机场的两个强收敛定理.高校应用数学学报(A 辑),2000 15(3):339-344.
[24]张立新,闻继威.B 值混合随机场的强大数律.数学年刊(A 辑),2001,22(2):205-216
[25] Utev,S.;Peligard,M.Maximal inequalities and an invariance principle for a class of weakly random varibales.J.Theoret.Probab.,2003,16(1):101-115.
[26] Peligard,M.;Gut,A.Almost sure results for a class of dependent random aribales. J.Theoret.Probab.,1999,12(1):87-104.
[27] }Zhang,L.X.Convergence rates in the strong law of nonstationary ρ *-mixing randomf ields.Acta.Math.Scientia.,2000,20(B):303-312.
[28] 杨善朝,α-混合序列和的强大数律及其应用,高校应用数学学报,1996(11,A,4),443-450
[29]佩特洛夫,B.B.,独立随机变量之和的极限定理,苏淳、黄可明译.中国科学技术大学出版社,1991.
[30]肖庆宪,郑祖康.随机过程序列部分和的收敛性[J].数学年刊 A辑,1999,20(2):177-182.
[31] Lin Z Y, Lu C R. The Limit Theory for Mixing Dependent Random Variables. Beijing: Science Press, 1997
[32] Peligrad,M.,Shao,Q.M. ,Estimation of the variance of partial sums for ρ -mixing random variables, J.Multi.Anal., 1995(52),140-157.
[33] Shao,Q.M., On the invariance principle for ρ -mixing sequences of random variables,Chinese Ann.Math.Ser., 1989, B 10 ,427-433
[34] Peligrad,M, Invariance principles for mixing sequences of random variables,Ann.Probab,1982,10,968-981
[35]Utev,S.A.,Central limit theorem for dependent random variables,Probability Theory and Mathematical Statistics(B.Grigel- uonis et al.,Eds),1990 , 2, 519-528
[36] Peligrad,M, The convergence of moments in the central limit theorem for ρ -mixing sequences of random variables, Proc.Amer. Math.Soc, 1987, 101,142-148
[37]Billingsley.p.Convergenceof Probability Measure,Wiley, New York.(1968)
[38] Utev.S.,Peligard,M.,Maximal inequalities an invariance principle for a class of weakly dependent random fields,J.Theor.Probab.,2003,16(1),101-115.
[39]王聪,张立新。 ρ *混合随机变量序列部分和的方差估计,应用概率统计,2005,12(4),419-430
[40] Peligard,M.,On the asymptotic normality of sequences of weak dependent random varibaels,J.Theoret.Probab.,1996(9),703-715.
[41] Bradley,R.C., On the spectral density and asymptotic normality of weakly dependent random fields,J.Theor.Probab., 1992(5), 355-373.
[42] Herrndorf,N., A functional central limit theorem for strongly mixing sequences of random variables,Z.Wahrsch.Verw.Gebiete,1985, 69,541-550.
[43] Ibragimov,I.A.and Linnik,Yu.V.,Independent and Stationary Sequences of Random Variables,Wolters,Groningen,1971.
[44] Bradley,R.C.and Bryc,W.,Multilinear forms and measures of dependence between random variables, J.Multivariate Anal.,1985,16, 335-367.
[45] Peligrad,M.and Utev,S.,Central limit theorem for linear processes, Ann.Probab.,1997,25:443-456.
[2] Kolmogorov, A.N.,Rozanov,U.A.,On the strong mixing conditions of a stationary Gaussian process ,Probab.Theory Appl.,1960(2),222-227. (In Russian)
[3] Peligrad M. Invariance principles for mixing sequenees of random variables,Ann.Prob.,1982 , 10 , 968-981
[4] PeligradM.On the central limit theorem for mixing sequences of random variables· Ann.Prob.,1987,15 , 1387-1394
[5] Lai T L.,Convergence rates and r- quick versions of the strong law for Stationary mixing sequences, Ann.Prob.,1977,5 , 693-706
[6] 邵启满,关于ρ -混合不变原理的注记.数学年刊,1988,9,409-412
[7] Peligrad M.,Convergence rates of the strong law of stationary mixing sequences, Z.Wahr. Verw. Gebiete,1985,70 , 307 一 314
[8]苏淳,关于一类强平稳 mixing 序列完全收敛性条件的探讨,应用概率统计,1988,4, 148 一 156
[9] 孔繁超.张明俊,关于ρ -混合序列的完全收敛性的注记.科学通报,1994,39(9),778-781
[10] 孔繁超, ρ -混合序列的完全收敛性,数学年刊(A)辑,1993,l(2),159-162
[11] 王晓明,非平稳- ρ 混合序列的完全收敛性,数学学报,1996,39(4),463-476
[12] Bradley,R.C.,Equivalent mixing conditions for random fields, Technical Report No .336: Center for Stochastic Processes ,Univ.of North Csrolina,Chapel Hill, 1990.
[13] Bradley,R.C., On the spectral density and asymptotic normality of weakly dependent random fields,J.Theor.Probab., 1992(5), 355-373.
[14] Peligard,M.,On the asymptotic normality of sequences of weak dependent random varibaels,J.Theoret.Probab.,1996(9),703-715.
[15] Peligrad,M.,Maximum of partial sums and an invariance principle for a class of weak dependent random variables,Proc.AMS.,1998,126(4), 181-1189.
[16] Utev.S.,Peligard,M.,Maximal inequalities an invariance principle for a class of weakly dependent random fields,J.Theor.Probab.,2003,16(1),101-115.
[17] Bradley,R.C.,Equivalent mixing conditions for random fields, Technical Report No .336: Center for Stochastic Processes ,Univ.of North Csrolina,Chapel Hill, 1990.
[18] Miller , C.Three theorems on ρ *mixing random fields.J.of Theoert.Probab.,1994,7(4):867-882.
[19]Miller , C.A CTL for periodgrams of a ρ *mixingrandom fields.Stoch.Proc.Appl.,1995,60: 313-330.
[20] Bryc,W.;Smolenski,W.Moment conditions for almost sure convergence of weakly cor-related random varibales.Porc.Amer.Math.Soc.,1993,119(2):629-635.
[21] 杨善朝.一类随机变量部分和的矩不等式及其应用.科学通报,1998,43(17):1836-182.7
[22] 张立新,B值强混合随机场的进一步矩不等式及强大数律,应用数学学报,2000,23(4), 518-525.
[23] 张立新,闻继威.弱负相关随机场的两个强收敛定理.高校应用数学学报(A 辑),2000 15(3):339-344.
[24]张立新,闻继威.B 值混合随机场的强大数律.数学年刊(A 辑),2001,22(2):205-216
[25] Utev,S.;Peligard,M.Maximal inequalities and an invariance principle for a class of weakly random varibales.J.Theoret.Probab.,2003,16(1):101-115.
[26] Peligard,M.;Gut,A.Almost sure results for a class of dependent random aribales. J.Theoret.Probab.,1999,12(1):87-104.
[27] }Zhang,L.X.Convergence rates in the strong law of nonstationary ρ *-mixing randomf ields.Acta.Math.Scientia.,2000,20(B):303-312.
[28] 杨善朝,α-混合序列和的强大数律及其应用,高校应用数学学报,1996(11,A,4),443-450
[29]佩特洛夫,B.B.,独立随机变量之和的极限定理,苏淳、黄可明译.中国科学技术大学出版社,1991.
[30]肖庆宪,郑祖康.随机过程序列部分和的收敛性[J].数学年刊 A辑,1999,20(2):177-182.
[31] Lin Z Y, Lu C R. The Limit Theory for Mixing Dependent Random Variables. Beijing: Science Press, 1997
[32] Peligrad,M.,Shao,Q.M. ,Estimation of the variance of partial sums for ρ -mixing random variables, J.Multi.Anal., 1995(52),140-157.
[33] Shao,Q.M., On the invariance principle for ρ -mixing sequences of random variables,Chinese Ann.Math.Ser., 1989, B 10 ,427-433
[34] Peligrad,M, Invariance principles for mixing sequences of random variables,Ann.Probab,1982,10,968-981
[35]Utev,S.A.,Central limit theorem for dependent random variables,Probability Theory and Mathematical Statistics(B.Grigel- uonis et al.,Eds),1990 , 2, 519-528
[36] Peligrad,M, The convergence of moments in the central limit theorem for ρ -mixing sequences of random variables, Proc.Amer. Math.Soc, 1987, 101,142-148
[37]Billingsley.p.Convergenceof Probability Measure,Wiley, New York.(1968)
[38] Utev.S.,Peligard,M.,Maximal inequalities an invariance principle for a class of weakly dependent random fields,J.Theor.Probab.,2003,16(1),101-115.
[39]王聪,张立新。 ρ *混合随机变量序列部分和的方差估计,应用概率统计,2005,12(4),419-430
[40] Peligard,M.,On the asymptotic normality of sequences of weak dependent random varibaels,J.Theoret.Probab.,1996(9),703-715.
[41] Bradley,R.C., On the spectral density and asymptotic normality of weakly dependent random fields,J.Theor.Probab., 1992(5), 355-373.
[42] Herrndorf,N., A functional central limit theorem for strongly mixing sequences of random variables,Z.Wahrsch.Verw.Gebiete,1985, 69,541-550.
[43] Ibragimov,I.A.and Linnik,Yu.V.,Independent and Stationary Sequences of Random Variables,Wolters,Groningen,1971.
[44] Bradley,R.C.and Bryc,W.,Multilinear forms and measures of dependence between random variables, J.Multivariate Anal.,1985,16, 335-367.
[45] Peligrad,M.and Utev,S.,Central limit theorem for linear processes, Ann.Probab.,1997,25:443-456.