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It will be shown and induced that the d-dimensional indices in the Banach spaces version conditions ∑n(E‖Xn‖p/|nα|p)<∞ are sufficient to yield limmin1≤j≤d(nj)→∞(1/|nα|)∑k≤n∏j=1d(1−(kj−1)/nj)Xk=0 a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.