Summary
The rate of convergence in the random strong law of large numbers for martingale differences is established. The results obtained generalize theorems given by R. Chen (1976) and I.A. Ahmad (1980).
References
Ahmad, I.A.: A remark on the SLLN for random partial sums. Z. Wahrscheinlichkeitstheor. Verw. Gebiete54, 119–124 (1980)
Babu, J.G., Ghosh, M.: A random functional central limit theorem for martingales. Acta Math. Sci. Hungar.27, 301–306 (1976)
Chen, R.: A remark on the tail probability of a distribution. J. Multivariate Analysis8, 328–333 (1978)
Chen, R.: A remark on the strong law of large numbers. Proc. Amer. Math. Soc.61, 112–116 (1976)
Csörgő, S., Rychlik, Z.: Rate of convergence in the strong law of large numbers. [To appear in Probability and Mathematical Statistics]
Egorov, V.A.: On the strong law of large numbers and the law of the iterated logarithm for sequences of independent random variables. Teor. Verojatnost. i Primenen.15, 520–527 (1970)
Feller, W.: An Introduction to Probability Theory and its Applications. Vol. I. New York: Wiley 1957
Fuk, D.H.: Some probability inequalities for martingales. Sibirskij Mat. Ž.14, 185–193 (1973)
Slivka, J., Severo, N.C.: On the strong law of large numbers. Proc. Am. Math. Soc.24, 729–734 (1970)
Siraždinov, S.H., Gafurov, M.V., Komekov, B.: Some remarks on the strong law of large numbers for sums of a random number of summands. Izv. Akad. Nauk. UzSSR Ser. Fiz. —Mat. Nauk, 28–34 (1978)
Szynal, D.: On almost complete convergence for the sums of a random of independent random variables. Bull. Acad. Polon. Sci., Ser. Math. Astronom. Phys.20, 571–574 (1972)
Teicher, H.: Generalized exponential bounds, iterated logarithm and strong laws. Z. Wahrscheinlichkeitstheor. Verw. Gebiete48, 293–307 (1979)
Wu, C.F.: A note on the convergence rate of the strong law of large numbers. Bull. Inst. Math. Acad. Sinica1, 121–124 (1973)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Łagodowski, Z.A., Rychlik, Z. Rate of convergence in the strong law of large numbers for martingales. Probab. Th. Rel. Fields 71, 467–476 (1986). https://doi.org/10.1007/BF01000217
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01000217
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Martingale Difference