Convergence of vector valued plurisubharmonic martingales with two indices (Q1209812)

In [\textit{A. V. Bukhvalov} and \textit{A. A. Danilevich}, Math. Notes, 31, 104-110 (1982; Zbl 0496.30029)], the notion of a (complex) Banach space \(X\) with the analytic Radon-Nikodym property was introduced. The analytic Radon-Nikodym property is equivalent to an a.s. convergence theorem for pluri-subharmonic martingales (cf. the above reference), like the classical Radon-Nikodym property is equivalent to the fulfillment of the Doob martingale convergence theorem. In this paper, among other characterizations and ramifications, it is proved that the analytic Radon-Nikodym property holds whenever the corresponding convergence theorem is valid for pluri-subharmonic martingales with two indices.