Two examples concerning martingales in Banach spaces

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 particulars for a Banach space X, having martingale type p for some p > 1 implies that X has martingale cotype q for some q < infinity. 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 of l(infinity)(2n) for 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.