@article{ECP1328,
author = {Sonja Cox and Mark Veraar},
title = {Some remarks on tangent martingale difference sequences in $L^1$-spaces},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {12},
year = {2007},
keywords = {tangent sequences; UMD Banach spaces; martingale difference sequences; decoupling inequalities; Davis decomposition},
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.},
pages = {no. 40, 421-433},
issn = {1083-589X},
doi = {10.1214/ECP.v12-1328},
url = {http://ecp.ejpecp.org/article/view/1328}}