Equivalence of Complete Convergence and Law of Large Numbers for B-Valued Random Elements

Abstract

Under some conditions on probability, this note discusses the equivalence between the complete convergence and the law of large number for B-valued independent random elements. The results of [10] become a simple corollary of the results here. At the same time, the author uses them to investigate the equivalence of strong and weak law of large numbers, and there exists an example to show that the conditions on probability are weaker.

Keywords

Complete convergence / B-valued random element / Law of large number / Equivalence / 60B12 / O211.4

References

[1] Azalarov T A, Volodin N A. Laws of large numbers for identically distributed Banach space valued random variables. Probab. Theor. Appl., 1981, 26: 573-580

[2] Bai Z D, Su C. The complete convergence for partial sums of iid random variables. Sci. Sinica (Ser.A), 1985, 28: 1261-1277

[3] de Acosta A. Inequaliyies of B-valued random vectors with applications to the strong law of large numbers. Ann. Probab., 1981, 9: 157-161

[4] Jain N C. Tail probabilities for sums of independent Banach space valued random variables. Z. Wahr. Verw. Gebiete, 1975, 33: 155-166

[5] Liang H Y, Gan S X, Ren Y F. Type of Banach space and complete convergence for sums of B-valued random element sequences. Acta Math. Sinica, 1997, 40: 449-456

[6] Romano, J. P. & Siegel, A. F., Counterexamples in probability and statistics, Wadsworth & Brooks, Cole advanced books & software, Montery, California, 1986.

[7] Wang X C, Rao M B, Yang X Y. Convergence rates on strong laws of large numbers for arrays of rowwise independent elements. Stochatic Anal. Appl., 1993, 11(1): 115-132

[8] Wu, Z. Q. & Wang, X. C., Probability in Banach space (in Chinese), Jilin University Press, China, 1990.

[9] Wu Z Q, Wang X C, Li D L. Some general results of the law of large numbers. Northeastern Math. J., 1987, 3(2): 228-238

[10] Yang X Y. Complete convergence of a class of independent B-valued random elements. Acta Math. Sinica, 1993, 36(6): 817-825