A local convergence theorem for a class of martingales with multidimensional time parameter is proved. This is accomplished by an elaboration of stopping time arguments used in the one-dimensional setting.
Richard F. Gundy. "Local Convergence of a Class of Martingales in Multidimensional Time." Ann. Probab. 8 (3) 607 - 614, June, 1980. https://doi.org/10.1214/aop/1176994731