On Convergence of Vector valued Pramarts and Subpramarts

Abstract

In [15] Millet and Sucheston introduced the notion of pramart and subpramart indexed by directed sets, generalizing that of martingale and submartingale, and studied their properties. In particular convergence theorems were proved. In this note we obtain convergence theorems for analogous Banach-valued processes.Let E be a Banach lattice with the Radon-Nikodym property. Let (Xt, , t ∊ J) be an E+-valued subpramart of class (d). Precise definitions are given below (Section 1).In [10] for J = N, Egghe proved a subpramart convergence theorem under the additional assumption that there is a subsequence {nk} ⊆ N such that converges weakly for each .