Electronic Communications in Probability

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

Sonja Cox and Mark Veraar

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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