This is an announcement for the paper "Some remarks on tangent martingale difference sequences in $L^1$-spaces" by Sonja Cox and Mark Veraar.

Abstract: Let $X$ be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ 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 \leq C_{p,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 $L^1$. We state several problems related to $(*)$ and other decoupling inequalities.

Archive classification: math.PR math.FA

Mathematics Subject Classification: 60B05; 46B09; 60G42

Citation: Electron. Commun. Probab. 12, 421-433, (2007)

The source file(s), tangent_arxiv.tex: 47306 bytes, is(are) stored in gzipped form as 0801.0695.gz with size 13kb. The corresponding postcript file has gzipped size 101kb.

Submitted from: mark@profsonline.nl

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