Abstract
We study complete convergence of martingale arrays under rather weak conditions. Our results considerably strengthen many of the results available in the literature. As a tool, we establish a martingale analogue of an inequality of Hoffman-Jørgensen which was earlier known only for independent random variables.
Similar content being viewed by others
REFERENCES
Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Prob. 1, 19–42.
Chow, Y. S. (1966). Some convergence theorems for independent random variables. Ann. Math. Stat. 37, 1482–1493.
Gut, A. (1993). Complete convergence. Preprint.
Hitczenko, P. (1990). Best constant in martingale version of Rosenthal's inequality. Ann. Prob. 18, 1656–1668.
Hoffman-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52, 159–186.
Hsu, P. L., and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. 33, 25–31.
Li, D., Rao, M. B., and Wang, X. (1992). Complete convergence of moving average processes. Stat. Prob. Lett. 14, 111–114.
Li, D., Rao, M. B., Jiang, T., and Wang, X. (1995). Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Prob. 8, 49–76.
Shao, Q. M. (1993). Complete convergence for α-mixing sequences. Stat. Prob. Lett. 16, 279–287.
Shao, Q. M. (1995). Maximal inequalities for partial sums of p-mixing sequences. Ann. Prob. 23, 938–947.
Yu, K. F. (1990). Complete convergence of weighted sums of martingale differences. J. Theor. Prob. 3, 339–347.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ghosal, S., Chandra, T.K. Complete Convergence of Martingale Arrays. Journal of Theoretical Probability 11, 621–631 (1998). https://doi.org/10.1023/A:1022646429754
Issue Date:
DOI: https://doi.org/10.1023/A:1022646429754