A. S. Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces

In this paper, we prove that under theF 4 conditions, anyL log+ L bounded two-parameter Banach spece valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if theF 4 condition is replaced by the weaker localF 4 condition.