Open Access
2007 Some remarks on tangent martingale difference sequences in $L^1$-spaces
Sonja Cox, Mark Veraar
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Electron. Commun. Probab. 12: 421-433 (2007). DOI: 10.1214/ECP.v12-1328

Abstract

Let $X$ be a Banach space. Suppose that for all $p\in $ 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.

Citation

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Sonja Cox. Mark Veraar. "Some remarks on tangent martingale difference sequences in $L^1$-spaces." Electron. Commun. Probab. 12 421 - 433, 2007. https://doi.org/10.1214/ECP.v12-1328

Information

Accepted: 29 October 2007; Published: 2007
First available in Project Euclid: 6 June 2016

zbMATH: 1137.46006
MathSciNet: MR2350579
Digital Object Identifier: 10.1214/ECP.v12-1328

Subjects:
Primary: 60B05
Secondary: 46B09 , 60G42

Keywords: Davis decomposition , Decoupling inequalities , Martingale difference sequences , tangent sequences , UMD Banach spaces

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