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Let {Xn;n  1} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides con- ditions under which the series P in Xi converges almost surely and the tail series {Tn = P i≫n Xi;n  1} satisfies supkn kTkk = OP(bn) and supkn kTkk Bn P → 0 for given fields of positive numbers {bn} and {Bn}. This result generalizes results of A. Rosalsky, J. Rosenblatt (7), (8) and S. H. Sung, A. I. Volodin (11).

ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES