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A Ratio Limit Theorem for Approximate Martingales

Published online by Cambridge University Press:  20 November 2018

Charles W. Lamb
Charles W. Lamb*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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It has been proved [3, p. 630] that the martingale convergence theorem obtained by Andersen and Jessen [1, p. 5] follows from the classical theory developed by Doob. By using some results of Yosida and Hewitt [9] on finitely additive set functions, Johansen and Karush [7] proved that the identification of the limit function as a derivative in the approach of Andersen and Jessen can be obtained in the general case. In this paper we sharpen the methods of Andersen and Jessen to obtain a ratio limit theorem for “approximate martingales”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 4 , August 1973 , pp. 772 - 779
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Andersen, E. S. and Jessen, B., Some limit theorems on set-functions, Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1948), 8 pp.Google Scholar
2. Chatterji, S. D., Differentiation along algebras, Manuscripta Math. 4 (1971), 213–224.Google Scholar
3. Doob, J. L., Stochastic processes (Wiley, New York, 1968).Google Scholar
4. Doob, J. L., Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458.Google Scholar
5. Doob, J. L., Discrete potential theory and boundaries, J. Math. Mech. 8 (1959), 433–458.Google Scholar
6. Hunt, G. A., Markov chains and Martin boundaries, Illinois J. Math. 4 (1960), 313–340.Google Scholar
7. Johansen, S. and Karush, J., On the semi-martingale convergence theorem. Ann. Math. Statist. 34 (1966), 690–694.Google Scholar
8. Sion, M., Introduction to the methods of real analysis (Holt, Rinehart and Winston, New York, 1969).Google Scholar
9. Yosida, K. and Hewitt, E., Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66.Google Scholar