Introduction. The subject of convergence of martingales of functions with values in a Banach
space was first treated by Scalora [17] and Chatterji [3] who independently showed that a …

The main result of Part I is Theorem 1. This is a maximal theorem for certain operators on
abstract L spaces, where 1? p< oo and E is a Banach space. This theorem contains as …

In recent years, the theory of martingales of real or complex random variables (for which see
DOOB [2], pp. 292--390) has been extended by various authors to random variables taking …

In recent years, several authors have considered various extensions of the martingale
convergence theorems of Doob [9] to the case where the random variables take values in a …

The purpose of the present paper is to establish some general condi tions for the existence
of integral representations of Banach-valued meas ures and indicate their relation to the …

The purpose of this article is to a give a general introduction to the problems arising from the
study of vector-valued martingales and their applications to scalar stochastic processes. We …

All Banach spaces in this paper will be understood to have complex scalars. More complete
definitions are in Section 1. A (complex-valued) bounded analytic function defined on the …

1. Introduction. The similarity between martingale convergence theorems and ergodic
theorems has been noticed for a long time. This similarity goes beyond the mere statements …

We shall give a necessary and sufficient condition for uniformly integrable martingales to
converge in the norm topology in a Banach function space. This yields a new …

By exploiting the natural setting of a convergence theorem of Burkholder, a direct and
elementary proof of the theorem is given. This proof is also new for the martingale …