Abstract
In this paper, we prove that under theF 4 conditions, anyL log+ L bounded two-parameter Banach spece valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if theF 4 condition is replaced by the weaker localF 4 condition.
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References
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Project supported by the National Natural Science Foundation of China and the State Education Commission Ph. D. Station Foundation
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Gan, S. A. S. Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces. Acta Mathematica Sinica 15, 187–196 (1999). https://doi.org/10.1007/BF02650662
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DOI: https://doi.org/10.1007/BF02650662
Keywords
- Two-parameter Banach space valued martingale
- A. S. convergence
- Radon-Nikodym property
- F 4 condition
- LocalF 4 condition