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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 485412, 12 pages
http://dx.doi.org/10.1155/2009/485412
Research Article

Strong Laws of Large Numbers for 𝔹-Valued Random Fields

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.

Linked References

  1. R. T. Smythe, “Strong laws of large numbers for r-dimensional arrays of random variables,” Annals of Probability, vol. 1, no. 1, pp. 164–170, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. I. Fazekas, “Marcinkiewicz strong law of large numbers for B-valued random variables with multidimensional indices,” in Statistics and Probability: Proceedings of the 3rd Pannonian Symposium on Mathematical Statistics, pp. 53–61, Visegrad, Hungary, September 1982. View at Zentralblatt MATH · View at MathSciNet
  3. R. T. Smythe, “Sums of independent random variables on partially ordered sets,” Annals of Probability, vol. 2, no. 5, pp. 906–917, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. Merzbach, “An introduction to the general theory of set-indexed martingales,” in Topics in Spatial Stochastic Processes, vol. 1802 of Lecture Notes in Mathematics, pp. 41–84, Springer, Berlin, Germany, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. Cairoli, “Une inégalité pour martingales à indices multiples et ses applications,” in Séminaire de Probabilités IV Université de Strasbourg, vol. 124 of Lecture Notes in Mathematics, pp. 1–27, Springer, Berlin, Germany, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J.-P. Gabriel, “Loi des grands nombres, séries et martingales à deux indices,” Comptes Rendus de l'Académie des Sciences. Série A, vol. 279, pp. 169–171, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. O. I. Klesov, “The Hájek-Rényi inequality for random fields and the strong law of large numbers,” Theory of Probability and Mathematical Statistics, vol. 22, pp. 63–71, 1980. View at Google Scholar · View at Zentralblatt MATH
  8. G. R. Shorack and R. T. Smythe, “Inequalities for max |Sk|/bk where kNr,” Proceedings of the American Mathematical Society, vol. 54, no. 1, pp. 331–336, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. J. Wichura, “Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters,” Annals of Mathematical Statistics, vol. 40, no. 2, pp. 681–687, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y. S. Chow, “A martingale inequality and the law of large numbers,” Proceedings of the American Mathematical Society, vol. 11, no. 1, pp. 107–111, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. C. Christofides and R. J. Serfling, “Maximal inequalities for multidimensionally indexed submartingale arrays,” The Annals of Probability, vol. 18, no. 2, pp. 630–641, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. J. Zimmerman, “Some sample function properties of the two-parameter Gaussian process,” Annals of Mathematical Statistics, vol. 43, no. 4, pp. 1235–1246, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. I. Martikainen, “Order of growth of a random field,” Mathematical Notes, vol. 39, no. 3, pp. 237–240, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. A. Woyczyński, “On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence,” Probability and Mathematical Statistics, vol. 1, no. 2, pp. 117–131, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. A. Woyczyński, “Geometry and martingales in Banach spaces. II: independent increments,” in Probability on Banach Spaces, vol. 4 of Advances in Probability and Related Topics, pp. 267–517, Marcel Dekker, New York, NY, USA, 1978. View at Google Scholar · View at MathSciNet
  16. H. D. Brunk, “The strong law of large numbers,” Duke Mathematical Journal, vol. 15, no. 1, pp. 181–195, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Yu. V. Prohorov, “On the strong law of large numbers,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 14, pp. 523–536, 1950. View at Google Scholar · View at MathSciNet
  18. A. Rosalsky and L. Van Thanh, “Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces,” Stochastic Analysis and Applications, vol. 24, no. 6, pp. 1097–1117, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. de Acosta, “Inequalities for B-valued random vectors with applications to the strong law of large numbers,” The Annals of Probability, vol. 9, no. 1, pp. 157–161, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet