Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 485412, 12 pages
doi:10.1155/2009/485412
Research Article

Strong Laws of Large Numbers for 𝔹-Valued Random Fields

Zbigniew A. Lagodowski

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka Street 38A, 20-618 Lublin, Poland

Received 30 October 2008; Revised 31 December 2008; Accepted 13 March 2009

Academic Editor: Stevo Stevic

Copyright © 2009 Zbigniew A. Lagodowski. 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

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for 𝔹-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.