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 Electronic Communications in Probability > Vol. 12 (2007) > Paper 41 open journal systems 


Some remarks on tangent martingale difference sequences in L1-spaces

Sonja Gisela Cox, TU Delft
Mark Christiaan Veraar, TU Delft


Abstract
Let X be a Banach space. Suppose that for all p in (1, ∞) a constant Cp,X depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has

E||f||p ≤ Cp,X E||g||p       (*).

This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L1. We state several problems related to (*) and other decoupling inequalities.


Full text: PDF

Pages: 421-433

Published on: October 29, 2007
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Electronic Communications in Probability. ISSN: 1083-589X