Premium
Two Examples Concerning Martingales in Banach Spaces
Author(s) -
Wenzel Jörg
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070500685x
Subject(s) - martingale (probability theory) , mathematics , local martingale , banach space , martingale difference sequence , pure mathematics , doob's martingale inequality , convexity , discrete mathematics , mathematical analysis , finance , economics
The analytic concepts of martingale type p and cotype q of a Banach space have an intimate relation with the geometric concepts of p ‐concavity and q ‐convexity of the space under consideration, as shown by pisier. In particular, for a banach space X , having martingale type p for some p > 1 implies that X has martingale cotype q for some q < ∞. The generalisation of these concepts to linear operators was studied by the author, and it turns out that the duality above only holds in a weaker form. An example is constructed showing that this duality result is best possible. So‐called random martingale unconditionality estimates, introduced by Garling as a decoupling of the unconditional martingale differences (UMD) inequality, are also examined. It is shown that the random martingale unconditionality constant ofl ∞ 2 nfor martingales of length n asymptotically behaves like n . This improves previous estimates by Geiss, who needed martingales of length 2 n to show this asymptotic. At the same time the order in the paper is the best that can be expected.