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
MathSciNet review: 0494472
Full-text PDF

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?)

  • [1] S. D. Chatterji, Martingale convergence and the Radon-Nikodým theorem in Banach spaces, Math. Scand. 22 (1968), 21-41. MR 39 #7645. MR 0246341 (39:7645)
  • [2] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [3] R. M. Dudley, Wiener functional as Itô integrals, Ann. Probability 5 (1977), 140-141. MR 0426151 (54:14097)
  • [4] H. P. McKean, Stochastic integrals, Academic Press, New York, 1969. MR 40 #947. MR 0247684 (40:947)

Similar Articles

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

Retrieve articles in all journals with MSC: 60G45

Additional Information

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

American Mathematical Society