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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Approximation of $ L\sp{1}$-bounded martingales by martingales of bounded variation

Authors: D. L. Burkholder and T. Shintani
Journal: Proc. Amer. Math. Soc. 72 (1978), 166-169
MSC: Primary 60G45
MathSciNet review: 0494472
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Abstract: If $ f = ({f_1},{f_2}, \ldots )$ is a real $ {L^1}$-bounded martingale and $ \varepsilon > 0$, then there is a martingale g of bounded variation satisfying $ {\left\Vert {f - g} \right\Vert _1} < \varepsilon $. The same result holds for X-valued martingales, where X is a Banach space, provided X has the Radon-Nikodým property. In fact, this characterizes Banach spaces having the Radon-Nikodým property. Theorem 1 identifies, for an arbitrary Banach space, the class of $ {L^1}$-bounded martingales that converge almost everywhere.

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Keywords: Martingale, bounded variation, Banach space, approximation, Radon-Nikodým property
Article copyright: © Copyright 1978 American Mathematical Society