随机序列的一些精确强极限定理
摘要
引文
[1] Acosta, A. de and Kuelbs, J. (1983) , Some results on the cluster set C({Sn/an}) and the LIL. Ann. Probab., 11: 102-122．
[2] Alam, K. and Saxena, K. M. L. (1981) , Positive dependence in multivariate distributions. Comm. Statist., A 10: 1183-1196．
[3] Alili, L. (1997) , On some byperbolic principal values of Brownian local times. Exponen-tial functionals and principal values related to Brownian motion, Biblioteca Rev. Mat. Iberoamericana, Madrid, 131-154．
[4] Anderson, T. W. (1955) , The integral of a symmetric unimodal funcion over a symmetric convex set and some probalbility inequalities. Proc. Am. Math. Soc., 6: 170-176．
[5] Barlow, M.T. and Yor, M. (1982) , Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times. J. Funct. anal, 49: 198-229．
[6] Baum, L. E. and Katz, M. (1965) , Convergence rates in the law of large numbers. Trans. Amer. Math. Soc., 120: 108-123．
[7] Bertoin, J. (1990) , Excursions of a BES0(d) and its drift term (0 < d < 1) . Probab. Theory Relat. Fields, 84: 231-250．
[8] Biane, P. and Yor, M. (1987) , Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math., 111: 23-101．
[9] Billingsley, P. (1968) , Convergence of Probability Measure. New York, Wiley.
[10] Borodin, A. N. (1985) , Distribution of the supremum of increments of Brownian local time, Problesm of the theory of probability distributions, IX. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Stedlov. (LOMI), 142: 6-24 (in Russian).
[11] Bovier, A. and Picco, P. (1993) , A law of the iterated logarithm for random geometric series. Ann. Probab., 21: 168-184．
[12] Chi, X. and Su, C. (1999) , An almost sure invariance principle for NA sequences. Preprint.
[13] Csaki, E., Csorgo, M., Foldes, A. and Shi, Z. (2000) , Increment sizes of the principal value of Brownian local time. Probab. Theory Relat. Fields, 117: 515-531．
[14] Csaki, E., Csorgo, M., Foldes, A. and Shi, Z. (2001) , Path properties of Cauchy's principal values related to local time. Studia Scientiarum Mathematicarum Hungarica, 38:149-169．
[15] Csaki, E., Konig, W. and Shi, Z. (1999) , An embedding for the Kesten-Spitzer random walk in random scenery. Stochast.Process.Appl., 82: 283-292．
[16] Csorgo, M. and Revesz, P. (1981) , Strong Approximations in Probability and Statistics. Akademiai Kiado. Budapest.
[17] Davis, J. A. (1968) , Convergence rates for probabilities of moderate diviations. Ann. Math. Statist. 39: 2016-2028．
[18] Dewan, I. and Prakasa Rao, B. L. S. (2001) , Asymptotic normality for U-statistics of associated random variables. J. Statist. Plan. Infe., 97: 201-225．
[19] Eagleson, G. K. (1979) , Orthogonal expansions and U-statistics. Austrai J. Statist., 21: 221-237．
[20] Einmahl, U. (1989) , The Darling-Erdos theorem for sums of i.i.d. random variables. Probab. Theory Relat. Fields, 82: 241-257．
[21] Einmahl, U.(1991) , On the almost sure behavior of sums of i.i.d. random variables in Hilbert space. Ann. Prob., 19: 1227-1263．
[22] Einmahl, U. (1993) , Toward a general Iaw of the iterated logarithm in banach space. Ann. Probab., 21: 2012-2045．
[23] Erdos, P. (1949) , On a theorem of Hsu and Robbins. Ann. Math. Statist. 20: 286-291．
[24] Erdos, P. (1950) , Remark on my paper "On a theorem of Hsu and Robbins." Ann. Math. Statist. 21: 138．
[25] Esary, J., Proschan, F. and Walkup, D. (1967) , Association of random variables with applications. Ann.Math.Statist., 38: 1466-1474．
[26] Feller, W. (1945) , The law of the iterated logarithm for identically distributed random variables. Ann. Math., 47: 631-638．
[27] Feller, W. (1968) , An Introduction to Probability Theory and Its Applications; Volume I: Thuird Edition. J. Wiley: New York.
[28] Gregory, G. G. (1977) , Large sample theory for U-statistics and tests of fits. Ann. Statist., 5: 110-123．
[29] Gut, A. and Spataru, A. (2000a), Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28: 1870-1883．
[30] Gut, A. and Spataru, A. (2000b), Precise asymptotics in the Baum-Katz and Davis Laws of large numbers. Jour. Math. Anal. Appl. 248: 233-246．
[31] Hall, P. (1979) , An invariance theorem for U-statistics. Stochast. Process. Appl, 9: 163-174．
[32] Hsu, P. L, and Robbins, H. (1947) , Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33: 25-31．
[33] Hu, Y. and Shi, Z. (1997) , An iterated logarithm law for Cauchy's principal value of Brow-nian local times. In: Exponential Functionals and Principal values Related to Brownian Motion (Yor, ed.), pp 131-154, Bibliotheca de la Revista Matematica Iberoamericana, Madrid.
[34] Ibragimov, I. A. and Linnik, Yu. V. (1971) , Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
[35] Ito, K. and McKean, H. P. (1965) , Diffusion processes and their sample paths. Springer, Berlin.
[36] Joag-Dev, K. and Proschan, F. (1983) , Negative association of random variables with applications. , Ann. Statist., 11: 286-295．
[37] Katz, M. (1963) , The probability in the tail of distribution. Ann. Math. Statist. 34: 312-318．
[38] Kesten, H. and Spitzer, F. (1979) , A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verew. Gebitte., 50: 5-25．
[39] Khoshnevisan, D. and Lewis, T. M. (1998) , A law fo the iterated logarithm for stable processes in random scenery. Stochast. Process. Appl., 74: 89-121．
[40] Klesov, O. and Rosalsky, A. (2001), A nonclassical law of the iterated logarithm for i.i.d. square integrable random variables. Stochast. Annl. Appl., 19: 627-641.
[41] Kolmogorov, A. N. and Rozanov, U. A. (1960), On the strong mixing conditions of stationary Gaussian process. Probab. Theory Appl. 2:222-227 (In Russian).
[42] Lehmann, E. L. (1966), Some concepts of bivariate dependence. Ann. Math. Starer., 37:1137-1153.
[43] 陆传荣，林正炎，1997，混合相依变量的极限理论，科学出版社．
[44] Matula, P. (1992), A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett., 15: 209-213.
[45] McKean, H. P. (1962), A Hlder condition for Brownian local time. J. Math. Kyoto Univ., 1: 195-201.
[46] Newman, C. M. (1984), Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. L. (Ed.), Inequalities in Statistics and Probability. IMS, Hayward, CA., pp. 127-140.
[47] Petrov, V. V., (1995), Limit Theorems of Probability Theory. Clarendon Press, Oxford.
[48] Picco, P. and Vares, M. E. (1994), A law of iterated logarithm for geometrically weighted martingale difference sequence. J. Theor. Probab., 7: 375-415.
[49] Révész, R. (1990), Random Walk in Random and Non-Random Environments. World Scientific, Singapore.
[50] Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion. Springer, Berlin, 3rd edition.
[51] Roussas, C. G. (1996), Exponential probability inequalities with some applications, In Ferguson, T. S., Shapley, L. S. and MacQueen, J. B. (eds.). Statistics, Probability and Game Theory. IMS, Hayward, CA., pp. 303-319.
[52] 邵启满，1988，关于p混合序列不变原理的注，数学年刊9A：409-412．
[53] 邵启满，1989，关于p混合序列的完全收敛性，数学学报32：377-393．
[54] Shao, Q. M. (1995), Maximal inequality for partical sums of p-mixing sequences. Ann. Prob. 23: 948-965.
[55] Shao, Q. M. and Su, C. (1999), The law of the iterated logarithm for negatively associated random variables. Stochast. Process. Appl., 86: 139-148.
[56] Shao, Q. M. (2000), A comparison theorem on maximum inequalities between negatively associated and independent random variables. J. Theoret. Probab., 13: 343-357.
[57] Spitzer, F. (1956), A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82: 323-339.
[58] Spǎtaru A. (1999), Precise asymptotics in Spitzer's law of large numbers. J. Theoret. Probab. 12: 811-819.
[59] Su, C., Zhao, L. C. and Wang, Y. B. (1997), The moment inequalities and weak convergence for negatively associated sequences. Science in China, 40A: 172-182.
[60] Su, C. and Wang, Y.B. (1998), Strong Convergence for IDNA esquences. Chinese J. Appl. Prob. Statist., 14: 131-140.
[61] Yor, M. (1997) , Eiponential Functionals and Principal values Related to Brownian Mo-tion. Bibliotheca de la Revista Matematica Iberoamericana, Madrid.
[62] Zhang, L. X. (1997) , Strong approximation theorems for geometrically weighted random series and their applications. Ann. Probab., 25: 1621-1635．
[63] Zhang, L. X. (1998) , The United Form of the Strong Invariance and the Complete Con-vergence. Acta Math. Sinica (A), 41 (6) : 1197-1210．
[64] Zhang, L. X. (2001a), A Strassen's law of the iterated logarithm for negatively associated random vectors. Stochast. Process. Appl., 95: 311-328．
[65] Zhang, L. X. (2001b), The weak convergence for functions of negatively associated random variables. J.Mult. Anal, 78: 272-298．
[66] Zhang, L. X. (2001c), The strong approximation for the Kesten-Spitzer random walk. Statis. Probab. Lett, 53: 21-26．
[67] Zhang, L. X. (2001d), The strong approximation for the general Kesten-Spitzer random walk in independent random scenery. Science in China, 44(A): 619-630．
[68] Zhang, L. X. and Wen, J. W. (2001) , A weak convergence for negatively associated fields. Statist. Probab. Lett., 53: 259-267．
[69] Zhang, L. X. (2002a), Precise rates in the law of the iterated logarithm. Preprint.
[70] Zhang, L. X. (2002b), Precise rate in the law of the iterated logarithm of NA sequences. Preprint.
[71] Zolotarev, V. M. (1961) , Concerning a certain probability problem. Theory Probab. Appl, 6: 201-204．
[2] Alam, K. and Saxena, K. M. L. (1981) , Positive dependence in multivariate distributions. Comm. Statist., A 10: 1183-1196．
[3] Alili, L. (1997) , On some byperbolic principal values of Brownian local times. Exponen-tial functionals and principal values related to Brownian motion, Biblioteca Rev. Mat. Iberoamericana, Madrid, 131-154．
[4] Anderson, T. W. (1955) , The integral of a symmetric unimodal funcion over a symmetric convex set and some probalbility inequalities. Proc. Am. Math. Soc., 6: 170-176．
[5] Barlow, M.T. and Yor, M. (1982) , Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times. J. Funct. anal, 49: 198-229．
[6] Baum, L. E. and Katz, M. (1965) , Convergence rates in the law of large numbers. Trans. Amer. Math. Soc., 120: 108-123．
[7] Bertoin, J. (1990) , Excursions of a BES0(d) and its drift term (0 < d < 1) . Probab. Theory Relat. Fields, 84: 231-250．
[8] Biane, P. and Yor, M. (1987) , Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math., 111: 23-101．
[9] Billingsley, P. (1968) , Convergence of Probability Measure. New York, Wiley.
[10] Borodin, A. N. (1985) , Distribution of the supremum of increments of Brownian local time, Problesm of the theory of probability distributions, IX. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Stedlov. (LOMI), 142: 6-24 (in Russian).
[11] Bovier, A. and Picco, P. (1993) , A law of the iterated logarithm for random geometric series. Ann. Probab., 21: 168-184．
[12] Chi, X. and Su, C. (1999) , An almost sure invariance principle for NA sequences. Preprint.
[13] Csaki, E., Csorgo, M., Foldes, A. and Shi, Z. (2000) , Increment sizes of the principal value of Brownian local time. Probab. Theory Relat. Fields, 117: 515-531．
[14] Csaki, E., Csorgo, M., Foldes, A. and Shi, Z. (2001) , Path properties of Cauchy's principal values related to local time. Studia Scientiarum Mathematicarum Hungarica, 38:149-169．
[15] Csaki, E., Konig, W. and Shi, Z. (1999) , An embedding for the Kesten-Spitzer random walk in random scenery. Stochast.Process.Appl., 82: 283-292．
[16] Csorgo, M. and Revesz, P. (1981) , Strong Approximations in Probability and Statistics. Akademiai Kiado. Budapest.
[17] Davis, J. A. (1968) , Convergence rates for probabilities of moderate diviations. Ann. Math. Statist. 39: 2016-2028．
[18] Dewan, I. and Prakasa Rao, B. L. S. (2001) , Asymptotic normality for U-statistics of associated random variables. J. Statist. Plan. Infe., 97: 201-225．
[19] Eagleson, G. K. (1979) , Orthogonal expansions and U-statistics. Austrai J. Statist., 21: 221-237．
[20] Einmahl, U. (1989) , The Darling-Erdos theorem for sums of i.i.d. random variables. Probab. Theory Relat. Fields, 82: 241-257．
[21] Einmahl, U.(1991) , On the almost sure behavior of sums of i.i.d. random variables in Hilbert space. Ann. Prob., 19: 1227-1263．
[22] Einmahl, U. (1993) , Toward a general Iaw of the iterated logarithm in banach space. Ann. Probab., 21: 2012-2045．
[23] Erdos, P. (1949) , On a theorem of Hsu and Robbins. Ann. Math. Statist. 20: 286-291．
[24] Erdos, P. (1950) , Remark on my paper "On a theorem of Hsu and Robbins." Ann. Math. Statist. 21: 138．
[25] Esary, J., Proschan, F. and Walkup, D. (1967) , Association of random variables with applications. Ann.Math.Statist., 38: 1466-1474．
[26] Feller, W. (1945) , The law of the iterated logarithm for identically distributed random variables. Ann. Math., 47: 631-638．
[27] Feller, W. (1968) , An Introduction to Probability Theory and Its Applications; Volume I: Thuird Edition. J. Wiley: New York.
[28] Gregory, G. G. (1977) , Large sample theory for U-statistics and tests of fits. Ann. Statist., 5: 110-123．
[29] Gut, A. and Spataru, A. (2000a), Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28: 1870-1883．
[30] Gut, A. and Spataru, A. (2000b), Precise asymptotics in the Baum-Katz and Davis Laws of large numbers. Jour. Math. Anal. Appl. 248: 233-246．
[31] Hall, P. (1979) , An invariance theorem for U-statistics. Stochast. Process. Appl, 9: 163-174．
[32] Hsu, P. L, and Robbins, H. (1947) , Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33: 25-31．
[33] Hu, Y. and Shi, Z. (1997) , An iterated logarithm law for Cauchy's principal value of Brow-nian local times. In: Exponential Functionals and Principal values Related to Brownian Motion (Yor, ed.), pp 131-154, Bibliotheca de la Revista Matematica Iberoamericana, Madrid.
[34] Ibragimov, I. A. and Linnik, Yu. V. (1971) , Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
[35] Ito, K. and McKean, H. P. (1965) , Diffusion processes and their sample paths. Springer, Berlin.
[36] Joag-Dev, K. and Proschan, F. (1983) , Negative association of random variables with applications. , Ann. Statist., 11: 286-295．
[37] Katz, M. (1963) , The probability in the tail of distribution. Ann. Math. Statist. 34: 312-318．
[38] Kesten, H. and Spitzer, F. (1979) , A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verew. Gebitte., 50: 5-25．
[39] Khoshnevisan, D. and Lewis, T. M. (1998) , A law fo the iterated logarithm for stable processes in random scenery. Stochast. Process. Appl., 74: 89-121．
[40] Klesov, O. and Rosalsky, A. (2001), A nonclassical law of the iterated logarithm for i.i.d. square integrable random variables. Stochast. Annl. Appl., 19: 627-641.
[41] Kolmogorov, A. N. and Rozanov, U. A. (1960), On the strong mixing conditions of stationary Gaussian process. Probab. Theory Appl. 2:222-227 (In Russian).
[42] Lehmann, E. L. (1966), Some concepts of bivariate dependence. Ann. Math. Starer., 37:1137-1153.
[43] 陆传荣，林正炎，1997，混合相依变量的极限理论，科学出版社．
[44] Matula, P. (1992), A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett., 15: 209-213.
[45] McKean, H. P. (1962), A Hlder condition for Brownian local time. J. Math. Kyoto Univ., 1: 195-201.
[46] Newman, C. M. (1984), Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. L. (Ed.), Inequalities in Statistics and Probability. IMS, Hayward, CA., pp. 127-140.
[47] Petrov, V. V., (1995), Limit Theorems of Probability Theory. Clarendon Press, Oxford.
[48] Picco, P. and Vares, M. E. (1994), A law of iterated logarithm for geometrically weighted martingale difference sequence. J. Theor. Probab., 7: 375-415.
[49] Révész, R. (1990), Random Walk in Random and Non-Random Environments. World Scientific, Singapore.
[50] Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion. Springer, Berlin, 3rd edition.
[51] Roussas, C. G. (1996), Exponential probability inequalities with some applications, In Ferguson, T. S., Shapley, L. S. and MacQueen, J. B. (eds.). Statistics, Probability and Game Theory. IMS, Hayward, CA., pp. 303-319.
[52] 邵启满，1988，关于p混合序列不变原理的注，数学年刊9A：409-412．
[53] 邵启满，1989，关于p混合序列的完全收敛性，数学学报32：377-393．
[54] Shao, Q. M. (1995), Maximal inequality for partical sums of p-mixing sequences. Ann. Prob. 23: 948-965.
[55] Shao, Q. M. and Su, C. (1999), The law of the iterated logarithm for negatively associated random variables. Stochast. Process. Appl., 86: 139-148.
[56] Shao, Q. M. (2000), A comparison theorem on maximum inequalities between negatively associated and independent random variables. J. Theoret. Probab., 13: 343-357.
[57] Spitzer, F. (1956), A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82: 323-339.
[58] Spǎtaru A. (1999), Precise asymptotics in Spitzer's law of large numbers. J. Theoret. Probab. 12: 811-819.
[59] Su, C., Zhao, L. C. and Wang, Y. B. (1997), The moment inequalities and weak convergence for negatively associated sequences. Science in China, 40A: 172-182.
[60] Su, C. and Wang, Y.B. (1998), Strong Convergence for IDNA esquences. Chinese J. Appl. Prob. Statist., 14: 131-140.
[61] Yor, M. (1997) , Eiponential Functionals and Principal values Related to Brownian Mo-tion. Bibliotheca de la Revista Matematica Iberoamericana, Madrid.
[62] Zhang, L. X. (1997) , Strong approximation theorems for geometrically weighted random series and their applications. Ann. Probab., 25: 1621-1635．
[63] Zhang, L. X. (1998) , The United Form of the Strong Invariance and the Complete Con-vergence. Acta Math. Sinica (A), 41 (6) : 1197-1210．
[64] Zhang, L. X. (2001a), A Strassen's law of the iterated logarithm for negatively associated random vectors. Stochast. Process. Appl., 95: 311-328．
[65] Zhang, L. X. (2001b), The weak convergence for functions of negatively associated random variables. J.Mult. Anal, 78: 272-298．
[66] Zhang, L. X. (2001c), The strong approximation for the Kesten-Spitzer random walk. Statis. Probab. Lett, 53: 21-26．
[67] Zhang, L. X. (2001d), The strong approximation for the general Kesten-Spitzer random walk in independent random scenery. Science in China, 44(A): 619-630．
[68] Zhang, L. X. and Wen, J. W. (2001) , A weak convergence for negatively associated fields. Statist. Probab. Lett., 53: 259-267．
[69] Zhang, L. X. (2002a), Precise rates in the law of the iterated logarithm. Preprint.
[70] Zhang, L. X. (2002b), Precise rate in the law of the iterated logarithm of NA sequences. Preprint.
[71] Zolotarev, V. M. (1961) , Concerning a certain probability problem. Theory Probab. Appl, 6: 201-204．