International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 86909, 15 pages
doi:10.1155/2007/86909
Research Article
Best Possible Sufficient Conditions for Strong Law of Large Numbers for Multi-Indexed Orthogonal Random Elements
Department of Business, National Open University, Lu Chow 247, Taipei, Taiwan
Received 26 April 2006; Revised 18 December 2006; Accepted 5 February 2007
Academic Editor: Andrew Rosalsky
Copyright © 2007 Kuo-Liang Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.