International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 86909, 15 pages
doi:10.1155/2007/86909

Best possible sufficient conditions for strong law of large numbers for multi-indexed orthogonal random elements

Abstract

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.