Two Examples Concerning Martingales in Banach Spaces

The analytic concepts of martingale type p and cotype q of aBanach space have an intimate relation with the geometric conceptsof p-concavity and q-convexity of the space under consideration,as shown by pisier. In particular, for a banach space X, havingmartingale type p for some p > 1 implies that X has martingalecotype q for some q < . The generalisation of these concepts to linear operators wasstudied by the author, and it turns out that the duality aboveonly holds in a weaker form. An example is constructed showingthat this duality result is best possible. So-called random martingale unconditionality estimates, introducedby Garling as a decoupling of the unconditional martingale differences(UMD) inequality, are also examined. It is shown that the random martingale unconditionality constantof for martingales of length n asymptotically behaves like n. This improves previous estimatesby Geiss, who needed martingales of length 2ⁿ to show this asymptotic.At the same time the order in the paper is the best that canbe expected.