Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1978-0494472-2
MathSciNet review: 0494472
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G45

Retrieve articles in all journals with MSC: 60G45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0494472-2
Keywords: Martingale, bounded variation, Banach space, approximation, Radon-Nikodým property
Article copyright: © Copyright 1978 American Mathematical Society