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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Martingales in the limit and amarts


Authors: G. A. Edgar and L. Sucheston
Journal: Proc. Amer. Math. Soc. 67 (1977), 315-320
MSC: Primary 60G45
MathSciNet review: 0488272
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Abstract: The notion of amart is compared to that of a martingale in the limit and game fairer with time. Every real-valued amart is a martingale in the limit. More generally, a Banach space E is finite-dimensional if and only if every E-valued amart is a martingale in the limit (or a game fairer with time). Several crucial properties possessed by amarts fail both for martingales in the limit and games fairer with time: the maximal inequality, the optional stopping theorem, the optional sampling theorem, the Riesz decomposition; therefore a general theory analogous to the amart theory cannot be based on the notion of a martingale in the limit. It is also observed that either the optional sampling theorem or a weak form of the Riesz decomposition must fail for any class of sequences of random variables strictly larger than the class of amarts.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0488272-6
PII: S 0002-9939(1977)0488272-6
Keywords: Martingale in the limit, amart, maximal inequality, optional sampling theorem, game fairer with time
Article copyright: © Copyright 1977 American Mathematical Society