CONDITIONS OF STRONG CONVERGENCE OF B-VALUED PRAMARTS

Let (\Omega, \mathscrF, P) be a probability space, D a directed set, and \left(\mathscrF_t\right)_t \in D a stochastic basis of \mathscrF. Let T be the set of all simple stdpping times with respect to \left(\mathscrF_t\right)_t \in D. In this note the following theorem is proved, which improves and extends the result of 5. Theorem. Let E be a Banach space with Radon-Nikodym property, and \left(x_t\right)_t \in D be an E-valued pramart. If there exists a cofinal subset \widetildeT of T such that the net \left(x_\tau\right)_\tau \in \tildeT is terminally uniformly integrable, then \left(x_\tau\right)_\tau \in T converges strongly stochastically (whence \left(x_\boldsymbolt\right)_\boldsymbolt \in \boldsymbolD converges strongly stochastically). If in addition \left(\mathscrF_\boldsymbolt\right)_\boldsymbolt \in \boldsymbolD satisfies Vitalj condition V, then \left(x_t\right)_t \in D converges strongly essentially.