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The Andersen-Jessen theorem revisited

Published online by Cambridge University Press:  24 October 2008

Klaus D. Schmidt
Klaus D. Schmidt
Affiliation:
Seminar für Statistik, Universität Mannheim, 6800 Mannheim, West Germany
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Abstract

For stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 351 - 361
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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