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Some inequalities for martingales and applications to the study of L1

Published online by Cambridge University Press:  24 October 2008

Leonard E. Dor
Leonard E. Dor
Affiliation:
Tel Aviv University, Israel
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Abstract

We obtain here estimates for the expectation of a restricted quadratic variation of L1 bounded (weak) martingales and positive submartingales, and for related functionals. These inequalities are then used to give a quantitative version of a recent result of D. J. Aldous and D. H. Fremlin concerning the L1 norms of expansions with respect to uniformly integrable L1-normalized martingale difference sequences. Another application relates precisely the degree of norm divergence of an L1-bounded martingale to the degree of non-uniform integrability of that martingale. A partial analogue of this result is proved for certain vector valued martingales with values in L1 which are related to the Radon–Nikodým property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 1 , January 1981 , pp. 135 - 148
Copyright
Copyright © Cambridge Philosophical Society 1981

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