A CONDITION OF UNIFORM INTEGRABILITY FOR B-VALUED MARTINGALES AND GEOMETRICAL PROPERTIES OF BANACH SPACES

We study, in this paper, some relationships of condition \lim _\lambda \rightarrow \infty \lambda P\left(S^2(f)>\lambda\right)=0 and uniform integrability of B-valued martingales （f_n, B_n）when the σ-algebra sequence （B_n） is regular. We prove that one of them implies another if the space which takes values in is q-convexifiable or p-smoothable, in particular, they are equivalent if the space is isomorphic to Hilbert space.