A.S. Convergence of Two-parameter Banach Space Valued Martingales and the Radon-Nikodym Property of Banach Spaces

摘要

In this paper, we prove that under the F_4 condition, any L log ~＋ L bounded two-parameter Banach space 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 the F_4 condition is replaced by the weaker local F_4 condition.