Electronic Communications in Probability

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

Sonja Cox and Mark Veraar

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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.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 40, 421-433.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224983

Digital Object Identifier
doi:10.1214/ECP.v12-1328

Mathematical Reviews number (MathSciNet)
MR2350579

Zentralblatt MATH identifier
1137.46006

Subjects
Primary: 60B05: Probability measures on topological spaces
Secondary: 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60G42: Martingales with discrete parameter

Keywords
tangent sequences UMD Banach spaces martingale difference sequences decoupling inequalities Davis decomposition

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Cox, Sonja; Veraar, Mark. Some remarks on tangent martingale difference sequences in $L^1$-spaces. Electron. Commun. Probab. 12 (2007), paper no. 40, 421--433. doi:10.1214/ECP.v12-1328. https://projecteuclid.org/euclid.ecp/1465224983


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