几类相依随机变量的强极限定理
摘要
引文
[1] Alam, K., Saxena, K.M.L. (1981), Positive dependence in multivariate distributions. Comm. Statist. A, 10: 1183-1196.
[2] Arnold, B.C., Villasenor, J.A. (1998), The asymptotic distribution of sums of records. Extremes, 1: 351-363.
[3] Balan, R.M. (2005), A strong invariance principle for associated random fields. Ann. Probab., 33: 823-840.
[4] Baldi, P., Rinott, Y. (1989), On normal approximation of distribution in terms of dependency graph. Ann. Probab., 17: 1646-1650.
[5] Baum, L.E., Katz, M. (1965), Convergence rates in the law of large numbers. Tran. Amer. Math. Soc., 120: 108-123.
[6] Bentkus, V., Gotze, F. (1996), The Berry-Esseen bound for Student's statistic. Ann. Probab., 24: 491-503.
[7] Berkes, I., Dehling, H. (1993), Some limit theorems in log density. Ann. Probab., 21: 1640-1670.
[8] Berkes, I., Csáki, E., Horváth, L. (1998), Almost sure central limit theorems under minimal conditions. Statist. Probab. Left., 37: 67-76.
[9] Berkes, I., Csáki, E. (2001), A universal result in almost sure central limit theory. Stochastic Process. Appl., 94: 105-134.
[10] Berry, A.C. (1941), The accuracy of the Gaussian approximation to the sum of independent variables. Trans. Amer. Math. Soc., 49: 122-136.
[11] Billingsley, P. (1968), Convergence of Probability Measures. Wiley, New York.
[12] Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987), Regular Variation. Cambridge University Press.
[13] Birkel, T. (1988a), On the convergence rate in the central limit theorem for associated processes. Ann. Probab., 16: 1685-1698.
[14] Birkel, T. (1988b), Moment bounds for associated sequences. Ann. Probab., 16: 1184-1193.
[15] Birkel, T. (1993), A functional central limit theorem for positively dependent random variables. J. Multivariate Anal., 44: 314-320.
[16] Brosamler, G.A. (1988), An almost everywhere central limit theorem. Math. Proc. Camb. Phil. Soc., 104: 561-574.
[17] Bulinski, A., Suquet, C. (2001), Normal approximation for quasi-associated random fields. Statist. Probab. Left., 54: 215-226.
[18] Chen, L.H.Y., Shao, Q.M. (2004), Normal approximation under local dependence. Ann. Probab., 32: 1985-2028.
[19] Chen, L.H.Y., Shao, Q.M. (2005), Stein's method for normal approximation. In An Introduction to Stein's method (editors: A.D. Barbour and L.H.Y. Chen), pp. 1-59. Lecture Notes Series, IMS, NUS, Vol. 4, Singapore.[20] Chen, R. (1978), A remark on the tail probaliblity of a distribution. J. Multivariate Anal., 8: 328-333.
[21] Chi, X.. Su, C. (1999), An almost sure invariance principle for NA sequences. Manuscript.
[22] Chow, Y.S. (1960), A martingale inequality and the law of large numbers. Proc. Amer. Math. Soc., 11: 107-111.
[23] Christofides, T.C. (2000), Maximal inequality for demimartingales and a strong law of large numbers. Statist. Probab. Lett., 50: 357-363.
[24] Csorgo, M., Révész, P. (1981), Strong Approximations in Probability and Statistics. Academic Press, New York.
[25] Csorgo, M., Szyszkowicz, B., Wang, Q. (2003a), Darling-Erdos theorem for self-normalized sums. Ann. Probab., 31: 676-692.
[26] Csorgo, M., Szyszkowicz, B., Wang, Q. (2003b), Donsker's theorem for self-normalized partial sums processes. Ann. Probab., 31: 1228-1240.
[27] Dabrowski, A.R., Dehling, H. (1988), A Berry-Esseen theorem and a functional law of the iterated logrithm for weakly associated random vector. Stochastic Process. Appl., 30: 277-289.
[28] 董志山，杨小云(2004)，NA及LNQD随机变量列的几乎处处中心极限定理．数学学报，47：593-600．
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[30] Erdos, P. (1949), On a theorem of Hsu and Robbins. Ann. Math. Statist., 20: 286-291.
[31] Esary, J.D., Proschan, F., Walkup, D.W. (1967), Association of random variables with applications. Ann. Math. Statist., 38: 1466-1474.
[32] Esseen, C.G. (1945), Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law. Acta Math., 77: 1-125.
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[34] Garsia, A. (1973), On a convex function inquality for submartingales. Ann. Probab., 1: 171-174.
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[36] Gnedenko, B.V., Kolmogorov, A.M. (1954), Limit Distributions for Sums of Independent Random Variables. Addison-wesley.
[37] Goldstein, L., Rinott, Y. (1996), Multivariate normal approximations by stein's method and size biased couplings. J. Appl. Probab., 33: 1-17.
[38] Griffin, P.S., Kuelbs, J.D. (1989), Self-normalized laws of the iterated logarithm. Ann. Probab., 17: 15711601.
[39] Gut, A., Spataru, A. (2000a), Precise asymptotics in the Baum-Katz and Davis law of large numbers. J. Math. Anal. Appl., 248: 233-246.[40] Gut, A., Spataru, A. (2000b), Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28: 1870-1883.
[41] Gut, A., Spataru, A. (2003), Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables. J. Multivariate Anal., 86: 398-422.
[42] Hajek, J., Rényi, A. (1955), A generalization of an inequality of kolomogorov. Acta. Math. Acad. Sci. Hung., 6: 281-284.
[43] Heyde, C.C. (1975), A supplement to the strong law of large numbers. J. Appl. Probab., 12: 173-175.
[44] Hsu, P.L., Robbins, H. (1947), Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A., 33: 25-31.
[45] 黄炜(2003)，随机序列的一些精确强极限定理．博士论文，浙江大学．
[46] Hurelbaatar, G. (1996), On the almost sure central limit theorem for φ-mixing random variables. Studia. Sci. Math. Hung., 31: 197-202.
[47] 蒋烨(2004)，随机序列的精确收敛性．博士论文，浙江大学．
[48] Jing, B.Y., Shao, Q.M., Wang, Q. (2003), Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab., 31: 2167-2215.
[49] Joag-Dev, K., Proschan, F. (1983), Negative association of random variables with applications. Ann. Statist., 11: 286-295.
[50] Khurelbaatar, G. (2001), Almost Sure Central Limit Theorems. Univ. of Cincinnati Ph.D. Dissertation.
[51] Khurelbaatar G., Rempala, G. (2006), A note on the almost sure limit theorem for the product of partial sums. Applied Mathematics Letters, 19: 191-196.
[52] Klesov, O., Rosalsky, A. (2001), A nonclassical law of the iterated logarithm for i.i.d, square integrable random variables. Stochastic Anal. Appl., 19: 627-641.
[53] Kochen, S., Stone, C. (1964), A note on the Borel-Cantelli lemma. Illinois J. Math., 8: 248-251.
[54] Lacey, M.T., Philipp, W. (1990), A note on the almost sure central limit theorem. Statist. Probab. Lett., 9: 201-205.
[55] Ledoux, M., Talagrand, M. (1991), Probability in Banach Space, Springer-Verlag, New York.
[56] Lehmann, E.L. (1966), Some concepts of dependence. Ann. Math. Statist., 37: 1137-1153.
[57] 李云霞(2005)，线性过程的若干极限理论及其应用．博士论文，浙江大学．
[58] 林正炎(1983)，U-统计量Berry-Esseen不等式的推广．应用数学学报，6：468-475．
[59] Lin, Z.Y. (1996), A self-normalized Chung-type law of the iterated logarithm. Theory. Probab. Appl., 41: 791-798.
[60] 陆传荣，林正炎(1997)，混合相依变量的极限理论．科学出版社，北京．[61] Lu, X., Qi, Y. (2004), A note on asymptotic distribution for products of sums. Statist. Probab. Lett., 68: 407-413.
[62] Matula, P. (1992), A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett., 15: 209-213.
[63] Newman, C.M. (1980), Normal fluctuations and the FKG inequalities. Comm. Math. Phys., 74: 119-128.
[64] Newman, C.M., Wright, A.L. (1981), An invariance principle for certain dependent sequences. Ann. Probab., 9: 671-675.
[65] Newman, C.M., Wright, A.L. (1982), Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Gebiete, 59: 361-371.
[66] Newman, C.M. (1984), Asympotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability(Tong, Y.L., ed., Institute of Mathematical Statistics, Hayward, CA), PP. 127-140.
[67] Pan, J.M. (1997), On the convergence rate in the central limit theorem for negatively associated sequences. Chinese J. Appl. Probab. Statist., 13: 183-192.
[68] 潘建敏，陆传荣(1998)，NA变量Esseen不等式的推广．科学通报，43：1144-1148．
[69] 庞天晓(2005)，随机变量序列的强极限性质．博士论文，浙江大学．
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[98] 王建峰，金敬森(2005)，关于多指标变量的Marcinkiewicz大数律和完全收敛性．数学物理学报(A辑)，25：734-743．
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[107] Zhang, L.X. (2000b), Central limit theorems for asymptotically negatively associated random fields. Acta Math. Sinica, New Ser., 16: 691-710.
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[109] Zhang, L.X., Wen, J.W. (2001b), A weak convergence for negatively associated fields. Statist. Probab. Lett., 53: 259-267.
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[111] Zhang, L.X. (2002a), Precise asymptotics in Chung's law of the iterated logarithm. Manuscript.
[112] Zhang, L.X. (2002b), Precise rates in the law of the logarithm under minimal conditions. Manuscript.
[113] 张立新(2004)，NA序列重对数律的几个极限定理．数学学报，47：541-552．
[114] 张立新，王江峰(2004)，两两NQD列的完全收敛性的一个注记．高校应用数学学报，19：203-208．
[2] Arnold, B.C., Villasenor, J.A. (1998), The asymptotic distribution of sums of records. Extremes, 1: 351-363.
[3] Balan, R.M. (2005), A strong invariance principle for associated random fields. Ann. Probab., 33: 823-840.
[4] Baldi, P., Rinott, Y. (1989), On normal approximation of distribution in terms of dependency graph. Ann. Probab., 17: 1646-1650.
[5] Baum, L.E., Katz, M. (1965), Convergence rates in the law of large numbers. Tran. Amer. Math. Soc., 120: 108-123.
[6] Bentkus, V., Gotze, F. (1996), The Berry-Esseen bound for Student's statistic. Ann. Probab., 24: 491-503.
[7] Berkes, I., Dehling, H. (1993), Some limit theorems in log density. Ann. Probab., 21: 1640-1670.
[8] Berkes, I., Csáki, E., Horváth, L. (1998), Almost sure central limit theorems under minimal conditions. Statist. Probab. Left., 37: 67-76.
[9] Berkes, I., Csáki, E. (2001), A universal result in almost sure central limit theory. Stochastic Process. Appl., 94: 105-134.
[10] Berry, A.C. (1941), The accuracy of the Gaussian approximation to the sum of independent variables. Trans. Amer. Math. Soc., 49: 122-136.
[11] Billingsley, P. (1968), Convergence of Probability Measures. Wiley, New York.
[12] Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987), Regular Variation. Cambridge University Press.
[13] Birkel, T. (1988a), On the convergence rate in the central limit theorem for associated processes. Ann. Probab., 16: 1685-1698.
[14] Birkel, T. (1988b), Moment bounds for associated sequences. Ann. Probab., 16: 1184-1193.
[15] Birkel, T. (1993), A functional central limit theorem for positively dependent random variables. J. Multivariate Anal., 44: 314-320.
[16] Brosamler, G.A. (1988), An almost everywhere central limit theorem. Math. Proc. Camb. Phil. Soc., 104: 561-574.
[17] Bulinski, A., Suquet, C. (2001), Normal approximation for quasi-associated random fields. Statist. Probab. Left., 54: 215-226.
[18] Chen, L.H.Y., Shao, Q.M. (2004), Normal approximation under local dependence. Ann. Probab., 32: 1985-2028.
[19] Chen, L.H.Y., Shao, Q.M. (2005), Stein's method for normal approximation. In An Introduction to Stein's method (editors: A.D. Barbour and L.H.Y. Chen), pp. 1-59. Lecture Notes Series, IMS, NUS, Vol. 4, Singapore.[20] Chen, R. (1978), A remark on the tail probaliblity of a distribution. J. Multivariate Anal., 8: 328-333.
[21] Chi, X.. Su, C. (1999), An almost sure invariance principle for NA sequences. Manuscript.
[22] Chow, Y.S. (1960), A martingale inequality and the law of large numbers. Proc. Amer. Math. Soc., 11: 107-111.
[23] Christofides, T.C. (2000), Maximal inequality for demimartingales and a strong law of large numbers. Statist. Probab. Lett., 50: 357-363.
[24] Csorgo, M., Révész, P. (1981), Strong Approximations in Probability and Statistics. Academic Press, New York.
[25] Csorgo, M., Szyszkowicz, B., Wang, Q. (2003a), Darling-Erdos theorem for self-normalized sums. Ann. Probab., 31: 676-692.
[26] Csorgo, M., Szyszkowicz, B., Wang, Q. (2003b), Donsker's theorem for self-normalized partial sums processes. Ann. Probab., 31: 1228-1240.
[27] Dabrowski, A.R., Dehling, H. (1988), A Berry-Esseen theorem and a functional law of the iterated logrithm for weakly associated random vector. Stochastic Process. Appl., 30: 277-289.
[28] 董志山，杨小云(2004)，NA及LNQD随机变量列的几乎处处中心极限定理．数学学报，47：593-600．
[29] Doob, J.L. (1953), Stochastic Processes. J. Wiley and Sons, New York.
[30] Erdos, P. (1949), On a theorem of Hsu and Robbins. Ann. Math. Statist., 20: 286-291.
[31] Esary, J.D., Proschan, F., Walkup, D.W. (1967), Association of random variables with applications. Ann. Math. Statist., 38: 1466-1474.
[32] Esseen, C.G. (1945), Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law. Acta Math., 77: 1-125.
[33] Feller, W. (1971), An Introduction to Probability Theory and its Applications 2, 2nd ed John Wiley, New York.
[34] Garsia, A. (1973), On a convex function inquality for submartingales. Ann. Probab., 1: 171-174.
[35] Giné, E., Gotze, F., Mason, D.M. (1997), When is the Student t-statistic asymptotically standand normal? Ann. Probab., 25: 1514-1531.
[36] Gnedenko, B.V., Kolmogorov, A.M. (1954), Limit Distributions for Sums of Independent Random Variables. Addison-wesley.
[37] Goldstein, L., Rinott, Y. (1996), Multivariate normal approximations by stein's method and size biased couplings. J. Appl. Probab., 33: 1-17.
[38] Griffin, P.S., Kuelbs, J.D. (1989), Self-normalized laws of the iterated logarithm. Ann. Probab., 17: 15711601.
[39] Gut, A., Spataru, A. (2000a), Precise asymptotics in the Baum-Katz and Davis law of large numbers. J. Math. Anal. Appl., 248: 233-246.[40] Gut, A., Spataru, A. (2000b), Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28: 1870-1883.
[41] Gut, A., Spataru, A. (2003), Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables. J. Multivariate Anal., 86: 398-422.
[42] Hajek, J., Rényi, A. (1955), A generalization of an inequality of kolomogorov. Acta. Math. Acad. Sci. Hung., 6: 281-284.
[43] Heyde, C.C. (1975), A supplement to the strong law of large numbers. J. Appl. Probab., 12: 173-175.
[44] Hsu, P.L., Robbins, H. (1947), Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A., 33: 25-31.
[45] 黄炜(2003)，随机序列的一些精确强极限定理．博士论文，浙江大学．
[46] Hurelbaatar, G. (1996), On the almost sure central limit theorem for φ-mixing random variables. Studia. Sci. Math. Hung., 31: 197-202.
[47] 蒋烨(2004)，随机序列的精确收敛性．博士论文，浙江大学．
[48] Jing, B.Y., Shao, Q.M., Wang, Q. (2003), Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab., 31: 2167-2215.
[49] Joag-Dev, K., Proschan, F. (1983), Negative association of random variables with applications. Ann. Statist., 11: 286-295.
[50] Khurelbaatar, G. (2001), Almost Sure Central Limit Theorems. Univ. of Cincinnati Ph.D. Dissertation.
[51] Khurelbaatar G., Rempala, G. (2006), A note on the almost sure limit theorem for the product of partial sums. Applied Mathematics Letters, 19: 191-196.
[52] Klesov, O., Rosalsky, A. (2001), A nonclassical law of the iterated logarithm for i.i.d, square integrable random variables. Stochastic Anal. Appl., 19: 627-641.
[53] Kochen, S., Stone, C. (1964), A note on the Borel-Cantelli lemma. Illinois J. Math., 8: 248-251.
[54] Lacey, M.T., Philipp, W. (1990), A note on the almost sure central limit theorem. Statist. Probab. Lett., 9: 201-205.
[55] Ledoux, M., Talagrand, M. (1991), Probability in Banach Space, Springer-Verlag, New York.
[56] Lehmann, E.L. (1966), Some concepts of dependence. Ann. Math. Statist., 37: 1137-1153.
[57] 李云霞(2005)，线性过程的若干极限理论及其应用．博士论文，浙江大学．
[58] 林正炎(1983)，U-统计量Berry-Esseen不等式的推广．应用数学学报，6：468-475．
[59] Lin, Z.Y. (1996), A self-normalized Chung-type law of the iterated logarithm. Theory. Probab. Appl., 41: 791-798.
[60] 陆传荣，林正炎(1997)，混合相依变量的极限理论．科学出版社，北京．[61] Lu, X., Qi, Y. (2004), A note on asymptotic distribution for products of sums. Statist. Probab. Lett., 68: 407-413.
[62] Matula, P. (1992), A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett., 15: 209-213.
[63] Newman, C.M. (1980), Normal fluctuations and the FKG inequalities. Comm. Math. Phys., 74: 119-128.
[64] Newman, C.M., Wright, A.L. (1981), An invariance principle for certain dependent sequences. Ann. Probab., 9: 671-675.
[65] Newman, C.M., Wright, A.L. (1982), Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Gebiete, 59: 361-371.
[66] Newman, C.M. (1984), Asympotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability(Tong, Y.L., ed., Institute of Mathematical Statistics, Hayward, CA), PP. 127-140.
[67] Pan, J.M. (1997), On the convergence rate in the central limit theorem for negatively associated sequences. Chinese J. Appl. Probab. Statist., 13: 183-192.
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