Abstract :
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 of l2n
∞ for martingales of
length n asymptotically behaves like n. This improves previous estimates by Geiss, who needed
martingales of length 2n to show this asymptotic. At the same time the order in the paper is the
best that can be expected.
