Abstract
We discuss the convergence rates in the law of logarithm for partial sums and randomly indexed partial sums of independent random variables in Banach space, and find the necessary and sufficient conditions on the convergence rates. The results of [1–3] for sums of i.i.d. real valued r.v.'s are extended; Yang's[4] result is generalized and the necessity part of Yang's result is also discussed; a conjecture for the i.i.d. real-valued r.v.'s of [5] is answered in Banach space.
Similar content being viewed by others
References
T.L. Lai. Limit Theorems for Deleyed Sums.Ann. Probab., 1974, 2: 432–440
J.A. Davis. Convergence Rates for Probabilities of Moderate Deviations.Ann. Math. Statist, 1968, 39: 2016–2028
A. Gut. Complete Convergence and Convergence Rates for Randomly Indexed Partial Sums with an Application to some First Passage Time.Acta. Math. Hung., 1983, 42 (3,4): 225–232
X.Y. Yang. Tail Probabilities for Sums ofB-valued Random Elements and Convergence Rates.Acta. Math. Appl. Sinica, 1992, 15: 322–332
A. Gut. Convergence Rates for Probabilities of Moderate Deviations for Sums of Random Variables with Multidimensional Indices.Ann. Probab., 1980, 8 (2): 298–313
P.L. Hsu, H. Robbins. Complete Convergence and the Law of Large Numbers.Proc. Nat. Acad. Sci., 1947, 33 (2): 25–31
Z.D. Bai, C. Su. The Complete Convergence for Partial Sums of i.i.d. Random Varibles.Sci. Sinica (Series A), 1985, 28: 1261–1277
T.C. Hu, N.C. Weber. On the Rate of Convergence in the Strong Law of Large Numbers for Arrays.Bull. Austral. Math. Soc., 1992, 45: 479–482
U. Einmahl. Stability Results and Strong Invariance Principles for Partial Sums of Banach Space Valued Random Variables.Ann. Probab., 1989, 17 (1): 333–352
M. Ledoux, M. Talagrand. Probability in Banach Spaces. Springer, Berlin, 1991
Q.M. Shao. On the Complete Convergence for Randomly Selected Sequences.Chinese Sci. Bull., 1990, 35: 93–98
N.C. Jain. Tail Probabilities for Sums of Independent Banach Space Valued Random Variables.Z. Wahr. Verw. Gebiete, 1975, 33: 155–166
A. de Acosta. Inequalities ofB-valued Random Vectors with Applications to the Strong Law of Large Numbers.Ann. Probab., 1981, 9: 157–161
H.Y. Liang, S.X. Gan, Y.F. Ren. Type of Banach Space and Complete Convergence for Sums ofB-valued Random Element Sequences.Acta. Math. Sinica, 1997, 40: 449–456
H.Y. Liang, C. Su. Complete Convergence for Weighted Sums of NA Sequences.Statist. & Probab. Lett., 1999, 45: 85–95
Y.B. Wang, X.G. Liu, C. Su. Equivalent Conditions of Complete Convergence for Independent Weighted Sums.Science in China (Series A), 1998, 41 (9): 939–949
L.E. Baum, M. Katz. Convergence Rates in the Law of Large Numbers.Trans. Amer. Soc., 1965, 120 (1): 108–123
Author information
Authors and Affiliations
Additional information
This work is supported by the National Natural Science Foundation of China (No. 10071081), the Natural Science Foundation of Education Department of Jiangsu Province and Science Fund of Tongji University.
Rights and permissions
About this article
Cite this article
Hanying, L., Chun, S. & Yuebao, W. Convergence rates in the law of logarithm of random elements. Acta Mathematicae Applicatae Sinica 17, 98–105 (2001). https://doi.org/10.1007/BF02669689
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02669689