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Some remarks on tangent martingale difference sequences in $L^1$-spaces

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Some remarks on tangent martingale difference sequences in $L^1$-spaces
2.	Creator	Author's name, affiliation, country	Sonja Gisela Cox; TU Delft
2.	Creator	Author's name, affiliation, country	Mark Christiaan Veraar; TU Delft
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	tangent sequences; UMD Banach spaces; martingale difference sequences; decoupling inequalities; Davis decomposition
3.	Subject	Subject classification	60B05; Secondary: 46B09, 60G42
4.	Description	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 $$\mathbb{E}\|f\|^p \leq C_{p,X} \mathbb{E}\|g\|^p \qquad (*).$$ 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	the Netherlands Organisation for Scientific Research (NWO) 639.032.201 and the Research Training Network MRTN-CT-2004-511953
7.	Date	(YYYY-MM-DD)	2007-10-29
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1328
10.	Identifier	Digital Object Identifier	10.1214/ECP.v12-1328
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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