一些随机变量序列部分和的大偏差定理
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摘要
近两个世纪以来，有关随机变量序列部分和的各种收敛性问题，如大数定律和中心极限定理等，一直是概率极限理论研究的主要问题，而关于随机变量序列部分和的大偏差却研究得很少。
设{X_n，n≥1}是定义在概率空间(Ω，F，μ)上的随机变量序列，S_n=sum from i=1 to n X_i，X_n∈L~p，n≥1，1≤p＜∞。如果{X_n，n≥1}是独立同分布(i.i.d.)的随机变量序列，则由弱大数定律知(?)μ(|S_n|＞nx)=0，x＞0。
更一般地，如果随机变量序列{X_n，n≥1}是强平稳的，则遍历性定理蕴含了上述结果仍然正确、有关μ(|S_n|＞nx)的收敛速度的问题，已经引起了很多学者的关注，在这中间，Nagaev(Theory Probab.Appl.10(1965)，214-235)得到了估计：μ(|S_n|＞n)=o(n~(1-p))，1≤p＜∞，X_i∈L~p，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)又证明了上述估计是最优的；如果{X_n，n≥1}是鞅差序列，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)证明了：如果sup_i E(e~(|X_i|))＜∞，则存在常数c＞0，使得μ(|S_n|＞n)≤e~(-cn~(1/3))，这个估计对于强平稳和遍历的鞅差序列来说是最优的；如果鞅差序列{X_n，n≥1}满足：X_i∈L~p，2≤p＜∞，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)又得到了估计：μ(|S_n|＞n)≤cn~(-p/2)，并且还证明了这个估计对于强平稳和遍历的鞅差序列来说也是最优的；Yulin Li(Statistics and Probability Letters，62(2003)，317-321)又将此结果推广到p∈(1，2]的情形，利用Burkholder不等式、C_r不等式和鞅的极大值不等式得到了估计：μ(|S_n|＞n)≤cn~(1-p)，在一定情况下，这个估计是最优的。
本文主要利用ρ混合序列、φ混合序列、(?)混合序列、(?)混合序列、NA序列、M-Z序列和线性过程序列的一些矩不等式，研究了它们的部分和序列S_n的大偏差定理，并且得到了与独立序列和鞅差序列类似的大偏差定理。
In recent two centuries, kinds of convergence properties for the partial sums of randomvariable sequences, such as strong law of large numbers and central limit theorem, havebeen the key subjects for probability limit theory research. But large deviations for thepartial sums of random variable sequences have seldom been studied.
Let {X_n, n≥1} be a random variable sequences defined on a fixed probability space(Ω,F,μ), and let S_n =sum from i=1 to n X_i, X_n∈L~p, n≥1, 1≤p＜∞. If {X_n, n≥1} isindependent and identically distributed(i.i.d.), the weak law of large numbers asserts that (?)μ(|S_n|＞nx)=0,x＞0.
More generally, if the sequence {X_n, n≥1} is stationary(in the strong sense), thenthe ergodic theorem asserts that the result is still true. In recent years, some authors havepaid much attentions to the problem of growth rate ofμ(|S_n|＞nx), for example, Na-gaev(Theory Probab.Appl.10(1965), 214-235) got the estimationμ(|S_n|＞n)= o(n~(1-p)) forX_i∈L~p, 1≤p＜∞, and Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)gave a simple proof that the estimate of Nagaev can't be improved; If {X_n, n≥1} is a mar-tingale difference sequence, Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159) proved that if sup_i E(e~(X_i))＜∞, then there exists a constant c＞0 such thatμ(|S_n|＞n)≤e~(-cn~(1/3)), this bound is optimal for the class of martingale difference sequenceswhich are also strictly stationary and ergodic; If the sequence {X_n, n≥1}is bounded inL~p, 2≤p＜∞, then Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)got the estimationμ(|S_n|＞n)≤cn~(-p/2) which is again optional for strictly station-ary and ergodic sequences of martingale difference; Yulin Li(Statistics and ProbabilityLetters, 62(2003), 317-321) generalized the result to the case for 1＜p≤2, by usingBurkholder's inequality, C_r-inequality and martingale maximal inequality, he obtainedμ(|S_n|＞n)≤cn~(1-p), these are optimal in a certain sense.
In the paper, we study the large deviations for the partial sums ofρ-mixing se- quence,φ-mixing sequence,(?)-mixing sequence,(?)-mixing sequence, NA sequence, M-Z-type sequence and Linear process sequence using some moment inequalities, and obtainthe similar results optimal upper bounds forμ(|S_n|＞n) as those for independent andidentically distributed sequence and martingale difference sequence.
设{X_n，n≥1}是定义在概率空间(Ω，F，μ)上的随机变量序列，S_n=sum from i=1 to n X_i，X_n∈L~p，n≥1，1≤p＜∞。如果{X_n，n≥1}是独立同分布(i.i.d.)的随机变量序列，则由弱大数定律知(?)μ(|S_n|＞nx)=0，x＞0。
更一般地，如果随机变量序列{X_n，n≥1}是强平稳的，则遍历性定理蕴含了上述结果仍然正确、有关μ(|S_n|＞nx)的收敛速度的问题，已经引起了很多学者的关注，在这中间，Nagaev(Theory Probab.Appl.10(1965)，214-235)得到了估计：μ(|S_n|＞n)=o(n~(1-p))，1≤p＜∞，X_i∈L~p，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)又证明了上述估计是最优的；如果{X_n，n≥1}是鞅差序列，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)证明了：如果sup_i E(e~(|X_i|))＜∞，则存在常数c＞0，使得μ(|S_n|＞n)≤e~(-cn~(1/3))，这个估计对于强平稳和遍历的鞅差序列来说是最优的；如果鞅差序列{X_n，n≥1}满足：X_i∈L~p，2≤p＜∞，Lesigne和Volny(Stochastic Process.Appl.96(2001)，143-159)又得到了估计：μ(|S_n|＞n)≤cn~(-p/2)，并且还证明了这个估计对于强平稳和遍历的鞅差序列来说也是最优的；Yulin Li(Statistics and Probability Letters，62(2003)，317-321)又将此结果推广到p∈(1，2]的情形，利用Burkholder不等式、C_r不等式和鞅的极大值不等式得到了估计：μ(|S_n|＞n)≤cn~(1-p)，在一定情况下，这个估计是最优的。
本文主要利用ρ混合序列、φ混合序列、(?)混合序列、(?)混合序列、NA序列、M-Z序列和线性过程序列的一些矩不等式，研究了它们的部分和序列S_n的大偏差定理，并且得到了与独立序列和鞅差序列类似的大偏差定理。
In recent two centuries, kinds of convergence properties for the partial sums of randomvariable sequences, such as strong law of large numbers and central limit theorem, havebeen the key subjects for probability limit theory research. But large deviations for thepartial sums of random variable sequences have seldom been studied.
Let {X_n, n≥1} be a random variable sequences defined on a fixed probability space(Ω,F,μ), and let S_n =sum from i=1 to n X_i, X_n∈L~p, n≥1, 1≤p＜∞. If {X_n, n≥1} isindependent and identically distributed(i.i.d.), the weak law of large numbers asserts that (?)μ(|S_n|＞nx)=0,x＞0.
More generally, if the sequence {X_n, n≥1} is stationary(in the strong sense), thenthe ergodic theorem asserts that the result is still true. In recent years, some authors havepaid much attentions to the problem of growth rate ofμ(|S_n|＞nx), for example, Na-gaev(Theory Probab.Appl.10(1965), 214-235) got the estimationμ(|S_n|＞n)= o(n~(1-p)) forX_i∈L~p, 1≤p＜∞, and Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)gave a simple proof that the estimate of Nagaev can't be improved; If {X_n, n≥1} is a mar-tingale difference sequence, Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159) proved that if sup_i E(e~(X_i))＜∞, then there exists a constant c＞0 such thatμ(|S_n|＞n)≤e~(-cn~(1/3)), this bound is optimal for the class of martingale difference sequenceswhich are also strictly stationary and ergodic; If the sequence {X_n, n≥1}is bounded inL~p, 2≤p＜∞, then Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)got the estimationμ(|S_n|＞n)≤cn~(-p/2) which is again optional for strictly station-ary and ergodic sequences of martingale difference; Yulin Li(Statistics and ProbabilityLetters, 62(2003), 317-321) generalized the result to the case for 1＜p≤2, by usingBurkholder's inequality, C_r-inequality and martingale maximal inequality, he obtainedμ(|S_n|＞n)≤cn~(1-p), these are optimal in a certain sense.
In the paper, we study the large deviations for the partial sums ofρ-mixing se- quence,φ-mixing sequence,(?)-mixing sequence,(?)-mixing sequence, NA sequence, M-Z-type sequence and Linear process sequence using some moment inequalities, and obtainthe similar results optimal upper bounds forμ(|S_n|＞n) as those for independent andidentically distributed sequence and martingale difference sequence.
引文
[1] Nagave, S.V. Some limit theorems for large deviations[J]. Theory Probab. Appl., 1965, 10: 214-235.
[2] Emmanuel Lesigne, Dalibor Volny. Large deviations for martingales[J]. Stochastic Processes and their Applications, 2001, 96: 143-159.
[3] Yulin Li. A martingale inequality and large deviations[J]. Statistics and Probability Letters, 2003, 62: 317-321.
[4] 匡继昌．常用不等式[M]．济南：山东科学技术出版社，2004：656-666．
[5] 缪柏其．概率论教程[M]．合肥：中国科学技术大学出版社，1998：9-61．
[6] Kolmogorov, A. N., Rozanov, U.A. On the strong mixing conditions of a stationary Gaussian process[J]. Probab. Theory Appl., 1960, 2: 222-227.
[7] Dobrushin, R. L. The central limit theorem for non-stationary Markov china[J]. Probab. Theory Appl., 1956, 1: 72-88.
[8] Peligrad, M. On the central limit theorem for ρ-mixing sequences of random variables[J]. Ann. Probab., 1987, 15: 1387-1394.
[9] Bradley, R. C. A central limit theorem for stationry ρ-mixing sequences with infinite variance[J]. Ann. Probab., 1988, 16: 313-332.
[10] 邵启满．关于ρ混合序列不变原理的注记[J]．数学年刊，1988，9A：409-412．
[11] 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．
[12] Shao, Q. M. On the invariance principle for p-mixing sequence with infinite variance[J]. Chin. Ann. Math., 1993, 14B: 27-42.
[13] 孔繁超，张俊．关于ρ混合序列完全收敛性的注记[J]．科学通报，1994，39：778-781．
[14] Shao, Q. M. Maximal inequality for partial sums of ρ-mixing sequences[J]. Ann. Probab., 1995, 23: 948-965.
[15] Peligrad, M., Shao, Q. M. Estimation of the variance of partial sums for p-mixing random variables[J]. J. Multivariate, 1995, 52: 140-157.
[16] Iosifescu, J. Limit theorem for (?)-mixing sequences[J]. A survey. Proc. 5th Conf. on Probab. Theory, Sept. 1-6, 1974, Brasov, Romania(Editura Acad. R. S. R., Bucuresti),1977: 51-57.
[17] Herrndorf, N. The invariance principle for (?)-mixing sequences[J]. Z.Wahrsch.verw Gebiete, 1983, 63: 97-108.
[18] Peligrad,M. An invariance principle for (?)-mixing sequences[J]. Ann. Probab., 1985, 13: 1304-1313.
[19] Utev, S. S. On the CLT for the series scheme of random variables with the (?)-mixing[J]. Probab. Theory Appl., 1990, 35: 110-117.
[20] Chen, D. Q. A uniform central limit theorem for non-uniform (?)-mixing random fields[J]. Ann. Probab., 1991, 19: 636-649.
[21] 杨善朝．混合序列矩不等式和非参数估计[J]．数学学报，1997，40(2)：271-279．
[22] 胡舒合．强大数定律的若干新结果[J]．数学学报，2003，46(6)：1123-1134．
[23] Borovkov, A.A.(Ed.). Advances In Probability Theorem: Limit Theorems For Sums of Random Variables[M]. Optimization Software, Inc. Publications Division, New York. 1985: 73-114.
[24] Bradley, R. C. Equivalent mixing conditions for random fields[R]. Technical Report No.336, Center for Stochastic Processes, Univ of North Carolina, Chapel Hill, 1990.
[25] 吴群英，林亮．(?)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．
[26] 吴群英．(?)混合序列的若干收敛性质[J]．工程数学学报，2001，18(3)：58-64．
[27] 吴群英．(?)混合序列加权和的完全收敛性和强收敛性[J]．应用数学，2002，15(1)：1-4．
[28] 吴群英．不同分布(?)混合序列的强收敛速度[J]．数学研究与评论，2004，24(1)：173-179
[29] Bryc W., Smolenski W. Moment conditions for almost sure convergence of weakly correlated random variables[J]. Proceeding of American Math Society, 1993, 119(2): 629-635.
[30] 杨善朝．一类随机变量部分和的矩不等式及其应用[J]．科学通报，1998，43(17)：1823-1827．
[31] 吴群英．混合序列的概率极限理论[M]．北京：科学出版社，2006：206-211．
[32] 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.
[33] Joag-Dev, K. and Proschan F. Negatively association of random variables with applications[J]. Ann. Statist., 1983, 11: 286-295.
[34] Matula, P. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics and Probability Letters. 1992,15: 209-213.
[35] 苏淳，赵林城，王岳宝．NA序列的矩不等式及弱收敛[J]．中国科学，A辑，1996，26：1091-1099．
[36] 苏淳，王岳宝．同分布NA序列的强收敛性[J]．应用概率统计，1998，14(2)：131-140．
[37] 王岳宝，周斌，苏淳．关于NA列部分和上升的阶[J]．应用概率统计，1998，14(2)：213-219
[38] Qi-Man Shao and Chun Su. The law of the iterated logarithm for negatively associated random variables[J]. Stochastic Process. Appl., 1999, 83: 139-148.
[39] 王小明．NA序列部分和的完全收敛性[J]．应用数学学报，1999，22(3)：407-412．
[40] Shao, Q. M. A comparison theorem on moment inequalities between negatively associated and independent random variables[J]. J. Theoretic. probab., 2000, 13: 343-356.
[41] 刘立新，吴荣．NA随机变量序列的强大数律和完全收敛[J]．应用概率统计，2001，17(3)：315-320．
[42] 王定成，苏淳，冷劲松．NA序列广义Jamison型加权和的几乎处处收敛性[J]．应用数学学报，2002，25(1)：77-87．
[43] Stout, W. F. Almost Sure Convergence[M]. New York: Academic Press, 1974: 30-150.
[44] Petrov，V．V．独立随机变量之和的极限定理[M]．苏淳、黄可明译．合肥：中国 科学技术大学出版社，1991：88-103．
[45] Tran, L. T., Roussas, G. G., Yakowitz, S. et al. Fixed-design regression for linear time series[J]. Ann. Statist., 1996, 24: 975-991.
[46] Hall, P., Heyde, C.C. Martingale Limit Theory and Its Application[M]. New York. London: Academic Press, INC. 1980.
[47] 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．
[48] Longnecker, M., Serfling, R. J. General moment and probability inequalities for the maximum partial sum[J]. Acta Math. Acad. Sci. Hungar., 1977, 30: 129-133.
[49] 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，1999：94-95．
[2] Emmanuel Lesigne, Dalibor Volny. Large deviations for martingales[J]. Stochastic Processes and their Applications, 2001, 96: 143-159.
[3] Yulin Li. A martingale inequality and large deviations[J]. Statistics and Probability Letters, 2003, 62: 317-321.
[4] 匡继昌．常用不等式[M]．济南：山东科学技术出版社，2004：656-666．
[5] 缪柏其．概率论教程[M]．合肥：中国科学技术大学出版社，1998：9-61．
[6] Kolmogorov, A. N., Rozanov, U.A. On the strong mixing conditions of a stationary Gaussian process[J]. Probab. Theory Appl., 1960, 2: 222-227.
[7] Dobrushin, R. L. The central limit theorem for non-stationary Markov china[J]. Probab. Theory Appl., 1956, 1: 72-88.
[8] Peligrad, M. On the central limit theorem for ρ-mixing sequences of random variables[J]. Ann. Probab., 1987, 15: 1387-1394.
[9] Bradley, R. C. A central limit theorem for stationry ρ-mixing sequences with infinite variance[J]. Ann. Probab., 1988, 16: 313-332.
[10] 邵启满．关于ρ混合序列不变原理的注记[J]．数学年刊，1988，9A：409-412．
[11] 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．
[12] Shao, Q. M. On the invariance principle for p-mixing sequence with infinite variance[J]. Chin. Ann. Math., 1993, 14B: 27-42.
[13] 孔繁超，张俊．关于ρ混合序列完全收敛性的注记[J]．科学通报，1994，39：778-781．
[14] Shao, Q. M. Maximal inequality for partial sums of ρ-mixing sequences[J]. Ann. Probab., 1995, 23: 948-965.
[15] Peligrad, M., Shao, Q. M. Estimation of the variance of partial sums for p-mixing random variables[J]. J. Multivariate, 1995, 52: 140-157.
[16] Iosifescu, J. Limit theorem for (?)-mixing sequences[J]. A survey. Proc. 5th Conf. on Probab. Theory, Sept. 1-6, 1974, Brasov, Romania(Editura Acad. R. S. R., Bucuresti),1977: 51-57.
[17] Herrndorf, N. The invariance principle for (?)-mixing sequences[J]. Z.Wahrsch.verw Gebiete, 1983, 63: 97-108.
[18] Peligrad,M. An invariance principle for (?)-mixing sequences[J]. Ann. Probab., 1985, 13: 1304-1313.
[19] Utev, S. S. On the CLT for the series scheme of random variables with the (?)-mixing[J]. Probab. Theory Appl., 1990, 35: 110-117.
[20] Chen, D. Q. A uniform central limit theorem for non-uniform (?)-mixing random fields[J]. Ann. Probab., 1991, 19: 636-649.
[21] 杨善朝．混合序列矩不等式和非参数估计[J]．数学学报，1997，40(2)：271-279．
[22] 胡舒合．强大数定律的若干新结果[J]．数学学报，2003，46(6)：1123-1134．
[23] Borovkov, A.A.(Ed.). Advances In Probability Theorem: Limit Theorems For Sums of Random Variables[M]. Optimization Software, Inc. Publications Division, New York. 1985: 73-114.
[24] Bradley, R. C. Equivalent mixing conditions for random fields[R]. Technical Report No.336, Center for Stochastic Processes, Univ of North Carolina, Chapel Hill, 1990.
[25] 吴群英，林亮．(?)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．
[26] 吴群英．(?)混合序列的若干收敛性质[J]．工程数学学报，2001，18(3)：58-64．
[27] 吴群英．(?)混合序列加权和的完全收敛性和强收敛性[J]．应用数学，2002，15(1)：1-4．
[28] 吴群英．不同分布(?)混合序列的强收敛速度[J]．数学研究与评论，2004，24(1)：173-179
[29] Bryc W., Smolenski W. Moment conditions for almost sure convergence of weakly correlated random variables[J]. Proceeding of American Math Society, 1993, 119(2): 629-635.
[30] 杨善朝．一类随机变量部分和的矩不等式及其应用[J]．科学通报，1998，43(17)：1823-1827．
[31] 吴群英．混合序列的概率极限理论[M]．北京：科学出版社，2006：206-211．
[32] 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.
[33] Joag-Dev, K. and Proschan F. Negatively association of random variables with applications[J]. Ann. Statist., 1983, 11: 286-295.
[34] Matula, P. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics and Probability Letters. 1992,15: 209-213.
[35] 苏淳，赵林城，王岳宝．NA序列的矩不等式及弱收敛[J]．中国科学，A辑，1996，26：1091-1099．
[36] 苏淳，王岳宝．同分布NA序列的强收敛性[J]．应用概率统计，1998，14(2)：131-140．
[37] 王岳宝，周斌，苏淳．关于NA列部分和上升的阶[J]．应用概率统计，1998，14(2)：213-219
[38] Qi-Man Shao and Chun Su. The law of the iterated logarithm for negatively associated random variables[J]. Stochastic Process. Appl., 1999, 83: 139-148.
[39] 王小明．NA序列部分和的完全收敛性[J]．应用数学学报，1999，22(3)：407-412．
[40] Shao, Q. M. A comparison theorem on moment inequalities between negatively associated and independent random variables[J]. J. Theoretic. probab., 2000, 13: 343-356.
[41] 刘立新，吴荣．NA随机变量序列的强大数律和完全收敛[J]．应用概率统计，2001，17(3)：315-320．
[42] 王定成，苏淳，冷劲松．NA序列广义Jamison型加权和的几乎处处收敛性[J]．应用数学学报，2002，25(1)：77-87．
[43] Stout, W. F. Almost Sure Convergence[M]. New York: Academic Press, 1974: 30-150.
[44] Petrov，V．V．独立随机变量之和的极限定理[M]．苏淳、黄可明译．合肥：中国 科学技术大学出版社，1991：88-103．
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