Some remarks on tangent martingale difference sequences in L1-spaces
Sonja Gisela Cox, TU Delft
Mark Christiaan Veraar, TU Delft
Abstract
Let X be a Banach space. Suppose that for all p in (1, ∞) a
constant Cp,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
E||f||p ≤ Cp,X E||g||p (*).
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 L1. We state several
problems related to (*) and other decoupling inequalities.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.