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Journal of Function Spaces and Applications
Volume 2013, Article ID 148249, 7 pages
http://dx.doi.org/10.1155/2013/148249
Research Article

Ideal Convergence of Random Variables

1Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791 112, India
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 31 May 2013; Accepted 7 September 2013

Academic Editor: Pankaj Jain

Copyright © 2013 B. Hazarika and S. A. Mohiuddine. 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. H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951. View at Google Scholar
  3. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 581–585, 2012. View at Google Scholar · View at MathSciNet
  7. S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,” Journal of Inequalities and Applications, vol. 2012, article 225, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces,” Advances in Difference Equations, vol. 2013, article 66, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 719729, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,” Applied Mathematics and Computation, vol. 219, no. 18, pp. 9821–9826, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  11. P. Kostyrko, T. Šalát, and W. Wilczyński, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2001. View at Google Scholar · View at MathSciNet
  12. K. Dems, “On I-Cauchy sequences,” Real Analysis Exchange, vol. 30, no. 1, pp. 123–128, 2005. View at Google Scholar · View at MathSciNet
  13. H. Cakalli and B. Hazarika, “Ideal quasi-Cauchy sequences,” Journal of Inequalities and Applications, vol. 2012, article 234, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “I and I*-convergence of double sequences,” Mathematica Slovaca, vol. 58, no. 5, pp. 605–620, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Hazarika and E. Savas, “Some I-convergent λ-summable difference sequence spaces of fuzzy real numbers defined by a sequence of Orlicz functions,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2986–2998, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 603–611, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. N. Šerstnev, “Random normed spaces: problems of completeness,” vol. 122, pp. 3–20, 1962. View at Google Scholar · View at MathSciNet
  19. A. N. Šerstnev, “On the concept of a stochastic normalized space,” Doklady Akademii Nauk SSSR, vol. 149, pp. 280–283, 1963. View at Google Scholar · View at MathSciNet
  20. C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a probabilistic normed space,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 91–98, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,” Mathematical Reports, vol. 12, no. 62, pp. 359–371, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Mursaleen and A. Alotaibi, “On I-convergence in random 2-normed spaces,” Mathematica Slovaca, vol. 61, no. 6, pp. 933–940, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  24. S. A. Mohiuddine, A. Alotaibi, and S. M. Alsulami, “Ideal convergence of double sequences in random 2-normed spaces,” Advances in Difference Equations, vol. 2012, article 149, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. M. Gürdal and I. Açık, “On I-Cauchy sequences in 2-normed spaces,” Mathematical Inequalities & Applications, vol. 11, no. 2, pp. 349–354, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. B. Hazarika, “On generalized difference ideal convergence in random 2-normed spaces,” Filomat, vol. 26, no. 6, pp. 1265–1274, 2012. View at Publisher · View at Google Scholar
  27. V. Kumar and B. L. Guillén, “On ideal convergence of double sequences in probabilistic normed spaces,” Acta Mathematica Sinica, vol. 28, no. 8, pp. 1689–1700, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. K. Kumar and V. Kumar, “On the I and I*-Cauchy sequences in probabilistic normed spaces,” Mathematical Sciences, vol. 2, no. 1, pp. 47–58, 2008. View at Google Scholar
  29. B. K. Lahiri and P. Das, “I and I*-convergence in topological spaces,” Mathematica Bohemica, vol. 130, no. 2, pp. 153–160, 2005. View at Google Scholar · View at MathSciNet
  30. S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence in fuzzy 2-normed space,” Journal of Computational Analysis and Applications, vol. 12, no. 4, pp. 787–798, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. U. Yamanci and M. Gürdal, “On lacunary ideal convergence in random 2-normed space,” Journal of Mathematics, vol. 2013, Article ID 868457, 8 pages, 2013. View at Publisher · View at Google Scholar
  34. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY, USA, 1983. View at MathSciNet
  35. D. A. Sibley, “A metric for weak convergence of distribution functions,” The Rocky Mountain Journal of Mathematics, vol. 1, no. 3, pp. 427–430, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, vol. 8 of Trends in Logic, Kluwer Academic, Dodrecht, The Netherlands, 2000. View at MathSciNet
  37. C. Alsina, B. Schweizer, and A. Sklar, “Continuity properties of probabilistic norms,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 446–452, 1997. View at Publisher · View at Google Scholar
  38. B. L. Guillén, J. A. R. Lallena, and C. Sempi, “Some classes of probabilistic normed spaces,” Rendiconti di Matematica e delle sue Applicazioni, vol. 17, no. 2, pp. 237–252, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet