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Convergent processes, projective systems of measures and martingale decompositions

Published online by Cambridge University Press:  18 May 2009

Louis H. Blake
Louis H. Blake
Affiliation:
Department of Mathematics, College of Staten Island, Cuny, Staten Island, New York 10301.
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The purpose of this paper is to show the equivalence of convergence, an associated projective system of measures and a martingale decomposition for a uniformly integrable stochastic process. Emphasis is placed on a direct juxtaposition of these concepts and on displaying underlying mechanisms.

The impact of the martingale convergence theorem on contemporary probability theory has been immense. Therein lies the reason for numerous generalizations of both the basic martingale convergence theorem and the martingale concept itself.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 20 , Issue 2 , July 1979 , pp. 119 - 124
Copyright
Copyright © Glasgow Mathematical Journal Trust 1979

References

REFERENCES

1.Krickeberg, K., Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313–337.CrossRefGoogle Scholar
2.Chatterji, S. D., Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21–41.CrossRefGoogle Scholar
3.Darst, R. B., Convergence of Radon-Nikodym derivatives and martingales given sigma lattices, Illinois J. Math. 21 (1977), 113–123.CrossRefGoogle Scholar
4.Fisk, D. L., Quasi-martingales, Trans. Amer. Math. Soc., 120 (1965), 369–389.CrossRefGoogle Scholar
5.Blake, L. H., A generalization of martingales and two consequent convergence theorems, Pacific J. Math. 35 (1970), 279–283.CrossRefGoogle Scholar
6.Blake, L. H., A note concerning the L1 convergence of a class of games which become fairer with time, Glasgow Math. J. 13 (1972), 39–41.CrossRefGoogle Scholar
7.Mucci, A. G., Limits for martingale-like sequences, Pacific J. Math. 48 (1973), 197–203.CrossRefGoogle Scholar
8.Mucci, A. G., Another martingale convergence theorem, Pacific J. Math. 64 (1976), 539–541.CrossRefGoogle Scholar
9.Austin, D. G., Edgar, G. A. and Tulcea, A. Ionesca, Pointwise convergence in terms of expectations, Zeit. fur Wahr. 30 (1974), 17–26.CrossRefGoogle Scholar
10.Edgar, G. A. and Sucheston, L., The Riesz decomposition for vector-valued amarts, Zeit. fur Wahr. 36 (1976), 85–92.Google Scholar
11.Baez-Duarte, L., Another look at the martingale theorem, J. Math. Anal. Appl. 23 (1968), 551–557.CrossRefGoogle Scholar
12.Lamb, C. W., A ratio limit theorem for approximate martingales, Canad. J. Math. 25 (1973), 772–779.CrossRefGoogle Scholar
13.Subramanian, R., On a generalization of martingales due to Blake, Pacific J. Math. 48 (1973), 275–278.CrossRefGoogle Scholar