This work addresses itself to the study of some potimal properties of Empirical Bayes estimators of the vector value parameters of the distributions. The goal is to provide some tools similar to those found for scalar parmeters. Relevant notions are defined and conditions of optimality for such estimators are stated. Utilizing the relations between - different modes of convergence, it has been established that componentwise asymptotic optimality is sufficient for vector asymptotic optimality. Thus, providing an easier way to check for asymptotic optimality.