Martingales in the limit and amarts

Edgar, G. A.; Sucheston, L.

doi:10.1090/S0002-9939-1977-0488272-6

Martingales in the limit and amarts
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by G. A. Edgar and L. Sucheston PDF

Proc. Amer. Math. Soc. 67 (1977), 315-320 Request permission

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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