相依变量的若干极限定理
摘要
本文是我在硕士阶段,在导师林正炎教授的悉心指导下完成的。全文分三章:
第一章,行内NA组列的一个完全收敛定理
自1947年Hsu和Robbins引进完全收敛性的概念以来,已有不少学者研究独立随机变量这方面的性质。Hu等人在1998年给出了关于独立组列的一个完全收敛定理(该定理不要求同分布条件)。之后有人对定理的条件作了探讨,做了一些相应的文章,但都限于独立的形式。本章把其中一些结论推广到行内为NA的情形,得到了如下几个定理:
定理 1.1.1 设{X_(nk),1≤k≤k_n,n≥1}是一个行内NA的组列,{c_n ,n≥1}是一个正的常数序列,满足。如果对于任意的ε>0和某个δ>0,下列条件成立:
(ⅱ) 存在J≥2,使得,
(ⅲ) 当n→∞时。
那么,对于任意的ε>0,有。
定理 1.1.2 设{X_(nk),1≤k≤k_n,n≥1)是一个行内NA的组列(如果k_n=∞),我们就假设级数几乎处处收敛),其中{k_n,n≥1}(?){1,2,…}∪{∞}。{c_n,n≥1}是一个正的常数序列,满足。如果对于任意的ε>0和某个δ>0,定理 1.1.1中条件(ⅰ),(ⅱ)仍然成立,条件(ⅲ)改为(ⅲ)′当n→∞时,那么,对于任意的ε>0,有。
但是定理 1.1.2的条件(ⅱ)和(ⅲ)′比较难以验证。对于均值为零的NA组列,我们有如下结果。
This thesis is finished under the guidance of my tutor, professor Lin Zhengyan, during my master of science. It consists of three chapters:Chapter 1 Complete convergence for arrays of rowwise negatively associated random variablesHsu and Robbins introduced the concept of complete convergence in 1947. From then on, many authors devote the study to the complete convergence for independent random variables. In 1998, Hu et al. presented a theorem concerning the complete convergence for an array of rowwise independent random variables. Then, some authors discussed the proof and the assumptions of the former theorem, and gave some relatively results. In this chapter, we extend some results to the rowwise negatively associated random variables:Theorem 0.1.1 Let {X_(nk),1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables, and {c_n, n≥1} a sequence of positive constants suchthat Σ_(n=1)~∞c_n=∞.Suppose that for all ε>0 and some δ>0,(ii) there exists J≥2 such thatThen for all ε>0Theorem 0.1.2 Let {X_(nk),1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables (if k_n =∞, we will assume that the series Σ_(k=1)~∞ X_(nk) converges a.s.) where {k_n,n≥1}(?){1,2,...}∪{∞}, and {c_n, n≥1} a sequence of positive constants such that Σ_(n=1)~∞c_n=∞. Suppose that for all ε>0 and some δ > 0, Conditions (i), (ii) andhold. Then
Mention that assumptions (ii) and (Hi)' of theorem 0.1.2 is somehow difficult to check. For a special case of mean zero array we can establish the following result: Theorem 0.1.3 Let {Xnk,\\} be an array of rowwise negativelyassociated random variables (if kn =00, we will assume that the series Xt-i^* converges a.s.) where {An,w>l}c{l,2,...}U{°°}, and {cn, n>\) a sequence of positive constants such that ^°°_lcn =00. Let q>(x) be a real function such that forsome S > 0: swpx>s x I 0:(II) there exists J>2 such that ￡I,c?(Ew￡M ^l))<00'(III) max^ I XL W\ X* I) l"> 0, as n -* qo .Then E!!lic-^max^lZHi^-*l>^<00 fora11 e>0-Chapter 2 Precise rates in the law of logarithm for positively associated random variablesSince Esary et al. introduced the concept of positively associated random variables, many authors have studied this concept and proved interesting results. Hsu and Robbins introduced the concept of complete convergence in 1947. Precise asymptotics are extensions of complete convergence. In this chapter, we extend the results of Fu KeAng^22' to the positively associated sequences:Theorem 0.2.1 Suppose that an = 0(1 / log n). Then for any b > -1, we haveandwhere //2(A+1) is the 2{b + l)th absolute moment of the standard normal distributions. Theorem 0.2.2 For any 6>-l,wehave— ^— /?>#? ■ i\2*+3
Theorem 0.23 Suppose that an = 0(1 / log ri). Then for any b > -1, we have//2(A+1)limff￡/(| ^ ^ (sH)ff^)where ^2(i+1) is the 2(Z> + l)th absolute moment of the standard normal distributions and 2t(￡) = {n :| 5? |> (e + an)\} be a sequence of 0?-mixing positive square integrable random variables, and it satisfies // = EX\ > 0 , E \ X\\\p< oo (p > 2), ￡"=y 2(2') < °o .Denote T^^'^-u), cr2n=ETn2. cr2n=ET^co as ?->?.Theorem 0.3.1 Let (Xki)j=l2..k,k = \,2,--- be a triangular array satisfies the following conditions: every row (Xkx,Xk2,---,Xkk) is an independent copy of (Xi,X2,---,Xk), but every row is independent. Denote Sk=Xkl+Xk2+-uk uuk u1T^logflWhere AT is a standard normal rv.
第一章,行内NA组列的一个完全收敛定理
自1947年Hsu和Robbins引进完全收敛性的概念以来,已有不少学者研究独立随机变量这方面的性质。Hu等人在1998年给出了关于独立组列的一个完全收敛定理(该定理不要求同分布条件)。之后有人对定理的条件作了探讨,做了一些相应的文章,但都限于独立的形式。本章把其中一些结论推广到行内为NA的情形,得到了如下几个定理:
定理 1.1.1 设{X_(nk),1≤k≤k_n,n≥1}是一个行内NA的组列,{c_n ,n≥1}是一个正的常数序列,满足。如果对于任意的ε>0和某个δ>0,下列条件成立:
(ⅱ) 存在J≥2,使得,
(ⅲ) 当n→∞时。
那么,对于任意的ε>0,有。
定理 1.1.2 设{X_(nk),1≤k≤k_n,n≥1)是一个行内NA的组列(如果k_n=∞),我们就假设级数几乎处处收敛),其中{k_n,n≥1}(?){1,2,…}∪{∞}。{c_n,n≥1}是一个正的常数序列,满足。如果对于任意的ε>0和某个δ>0,定理 1.1.1中条件(ⅰ),(ⅱ)仍然成立,条件(ⅲ)改为(ⅲ)′当n→∞时,那么,对于任意的ε>0,有。
但是定理 1.1.2的条件(ⅱ)和(ⅲ)′比较难以验证。对于均值为零的NA组列,我们有如下结果。
This thesis is finished under the guidance of my tutor, professor Lin Zhengyan, during my master of science. It consists of three chapters:Chapter 1 Complete convergence for arrays of rowwise negatively associated random variablesHsu and Robbins introduced the concept of complete convergence in 1947. From then on, many authors devote the study to the complete convergence for independent random variables. In 1998, Hu et al. presented a theorem concerning the complete convergence for an array of rowwise independent random variables. Then, some authors discussed the proof and the assumptions of the former theorem, and gave some relatively results. In this chapter, we extend some results to the rowwise negatively associated random variables:Theorem 0.1.1 Let {X_(nk),1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables, and {c_n, n≥1} a sequence of positive constants suchthat Σ_(n=1)~∞c_n=∞.Suppose that for all ε>0 and some δ>0,(ii) there exists J≥2 such thatThen for all ε>0Theorem 0.1.2 Let {X_(nk),1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables (if k_n =∞, we will assume that the series Σ_(k=1)~∞ X_(nk) converges a.s.) where {k_n,n≥1}(?){1,2,...}∪{∞}, and {c_n, n≥1} a sequence of positive constants such that Σ_(n=1)~∞c_n=∞. Suppose that for all ε>0 and some δ > 0, Conditions (i), (ii) andhold. Then
Mention that assumptions (ii) and (Hi)' of theorem 0.1.2 is somehow difficult to check. For a special case of mean zero array we can establish the following result: Theorem 0.1.3 Let {Xnk,\
Theorem 0.23 Suppose that an = 0(1 / log ri). Then for any b > -1, we have//2(A+1)limff￡/(| ^ ^ (sH)ff^)where ^2(i+1) is the 2(Z> + l)th absolute moment of the standard normal distributions and 2t(￡) = {n :| 5? |> (e + an)
引文
[1] Hu T.C., Szynal D., Volodin A.I., A note on complete convergence for arrays, Statist. Probab. Lett., 1998, 38: 27-31.
[2] Hu T.C., Volodin A.I., Addendum to "A note on complete convergence for arrays", Statist. Probab. Lett., 2000, 47:209-211.
[3] Hu T.C., Cabrera M.O., et., Complete convergence for arrays of rowwise independent random variables, Commun. Korean Math. Soc., 2003, 18, No.2, 375-383.
[4] Kuczmaszewska A., On some conditions for complete convergence for arrays, Statist. Probab. Lett., 2004, 66: 399-405.
[5] Sung S.H., Volodin A.I., Hu T.C., More on complete convergence for arrays, Statist. Probab. Lett., 2005, 71: 303-311.
[6] Chen P.Y., Gan S.X., Remark on complete convergence for arrays, Statist. Probab. Lett., (In Press)
[7] Yang X.R., A Hoffmann-Jorgensen inequality of NA random variables with applications to the convergence rate and the logarithm law, (manuscript).
[8] Shao Q.M., A comparison theorem on moment inequalities between negatively associated and independent variables, Journal of Theoretical Probability, 2000, Vol.13, No.2, 343-356.
[9] Etemadi N., On some classical results in probability theory, Sankhyā, Ser., 1985, A 47: 215-221.
[10] Billingsley P., Convergence of probability measures, New York: Wiley, 1995.
[11] Petrov V.V., Almost sure convergence, New York: Academic Press, 1995.
[12] Gut A., Spǎtaru A., Precise asymptotics in the Baum-Katz and Davis law of large numbers, d. Math. Anal. Appl., 2000, 248: 233-246.
[13] Pang T.X., Lin Z.Y., Precise rates in the law of logarithm for i.i.d random variables, Comput. Math. Appl., 2005, 49 (7-8): 997-1010.
[14] Thomas E. Wood, A Berry-Esseen theorem for associated random variables, Ann. Probab., 1983, 11: 1042-1047.
[15] Newman C.M., Normal fluctuations and the FKG inequalities, Comm. Math. Phys., 1980, 74: 119-128.
[16] Newman C.M., Wright A., An invariance principle for certain dependent sequences, Ann. Probab., 1981, 9: 671-675.
[17] Esary J., Proschan F., Walkup D., Association of random variables with applications, Ann. Math. Statist, 1967, 38:1466-1474.
[18] Dabrowski A.R., A functional law of the iterated logarithm for associated sequences, Statist. Probab. Lett., 1985, 3: 209-213.
[19] Birkel T., Moment bounds for associated sequences, Ann. Probab., 1988, 16: 1184-1193.
[20] Birkel T., A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett., 1989, 7: 17-20.
[21] Lin Z.Y., An invariance principle for associated random variables, Chinese. Ann. Math(Series A), 1996, 17: 487-494.
[22] Fu K.A., Zhang L.X., Pricise rates in the law of the logarithm for negtively associated random variables, (Manuscript).
[23] Arnold B.C., Villasenor J.A., The asymptotic distribution of sums of records, Extremes, 1998, 1:3, 351-363.
[24] Rempala G., Wesotowski J., Asymptotics for products of sums and U-Statistics, Elect. Comm. In Probab., 2002, 7: 47-54.
[25] Rempata G., Wesolowski J., Asymptotics for products of independent sums with an application to Wishart determinants, Statist. Probab. Lett.,2005, 74:129-138.
[26] Qi Y.C., Limit distributions for products of sums, Statist. Probab. Lett., 2003, 62: 93-100.
[27] Qi Y.C., A note on asymptotic distribution of products of sums, Statist. Probab. Lett., 2004, 68: 407-413.
[28] Lin Z.Y., Lu C.R., Limit Theory for Mixing Dependent Random Variables, Science Press/Kluwer Academic Publishers, 1996.
[29] 杨善朝,PA序列部分和的完全收敛性,应用概率统计,2001,17:197-202.
[30] 邵启满,相伴随机场的Berry-Esseen不等式和不变原理,应用概率统计,1989,5:1-8.
[31] 于浩,相伴随机变量的重对数律,数学学报,1986,29:507-511.
[32] 张立新,NA序列重对数律的几个极限定理,数学学报,2004,47:541-552.
[33] 林正炎,陆传荣,苏中根,概率极限理论基础,高等教育出版社,1999.
[34] 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),已录用.
[2] Hu T.C., Volodin A.I., Addendum to "A note on complete convergence for arrays", Statist. Probab. Lett., 2000, 47:209-211.
[3] Hu T.C., Cabrera M.O., et., Complete convergence for arrays of rowwise independent random variables, Commun. Korean Math. Soc., 2003, 18, No.2, 375-383.
[4] Kuczmaszewska A., On some conditions for complete convergence for arrays, Statist. Probab. Lett., 2004, 66: 399-405.
[5] Sung S.H., Volodin A.I., Hu T.C., More on complete convergence for arrays, Statist. Probab. Lett., 2005, 71: 303-311.
[6] Chen P.Y., Gan S.X., Remark on complete convergence for arrays, Statist. Probab. Lett., (In Press)
[7] Yang X.R., A Hoffmann-Jorgensen inequality of NA random variables with applications to the convergence rate and the logarithm law, (manuscript).
[8] Shao Q.M., A comparison theorem on moment inequalities between negatively associated and independent variables, Journal of Theoretical Probability, 2000, Vol.13, No.2, 343-356.
[9] Etemadi N., On some classical results in probability theory, Sankhyā, Ser., 1985, A 47: 215-221.
[10] Billingsley P., Convergence of probability measures, New York: Wiley, 1995.
[11] Petrov V.V., Almost sure convergence, New York: Academic Press, 1995.
[12] Gut A., Spǎtaru A., Precise asymptotics in the Baum-Katz and Davis law of large numbers, d. Math. Anal. Appl., 2000, 248: 233-246.
[13] Pang T.X., Lin Z.Y., Precise rates in the law of logarithm for i.i.d random variables, Comput. Math. Appl., 2005, 49 (7-8): 997-1010.
[14] Thomas E. Wood, A Berry-Esseen theorem for associated random variables, Ann. Probab., 1983, 11: 1042-1047.
[15] Newman C.M., Normal fluctuations and the FKG inequalities, Comm. Math. Phys., 1980, 74: 119-128.
[16] Newman C.M., Wright A., An invariance principle for certain dependent sequences, Ann. Probab., 1981, 9: 671-675.
[17] Esary J., Proschan F., Walkup D., Association of random variables with applications, Ann. Math. Statist, 1967, 38:1466-1474.
[18] Dabrowski A.R., A functional law of the iterated logarithm for associated sequences, Statist. Probab. Lett., 1985, 3: 209-213.
[19] Birkel T., Moment bounds for associated sequences, Ann. Probab., 1988, 16: 1184-1193.
[20] Birkel T., A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett., 1989, 7: 17-20.
[21] Lin Z.Y., An invariance principle for associated random variables, Chinese. Ann. Math(Series A), 1996, 17: 487-494.
[22] Fu K.A., Zhang L.X., Pricise rates in the law of the logarithm for negtively associated random variables, (Manuscript).
[23] Arnold B.C., Villasenor J.A., The asymptotic distribution of sums of records, Extremes, 1998, 1:3, 351-363.
[24] Rempala G., Wesotowski J., Asymptotics for products of sums and U-Statistics, Elect. Comm. In Probab., 2002, 7: 47-54.
[25] Rempata G., Wesolowski J., Asymptotics for products of independent sums with an application to Wishart determinants, Statist. Probab. Lett.,2005, 74:129-138.
[26] Qi Y.C., Limit distributions for products of sums, Statist. Probab. Lett., 2003, 62: 93-100.
[27] Qi Y.C., A note on asymptotic distribution of products of sums, Statist. Probab. Lett., 2004, 68: 407-413.
[28] Lin Z.Y., Lu C.R., Limit Theory for Mixing Dependent Random Variables, Science Press/Kluwer Academic Publishers, 1996.
[29] 杨善朝,PA序列部分和的完全收敛性,应用概率统计,2001,17:197-202.
[30] 邵启满,相伴随机场的Berry-Esseen不等式和不变原理,应用概率统计,1989,5:1-8.
[31] 于浩,相伴随机变量的重对数律,数学学报,1986,29:507-511.
[32] 张立新,NA序列重对数律的几个极限定理,数学学报,2004,47:541-552.
[33] 林正炎,陆传荣,苏中根,概率极限理论基础,高等教育出版社,1999.
[34] 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),已录用.