Afrika Matematika

, Volume 25, Issue 3, pp 681–692

\(\mathcal{I }\)-statistical convergence of a sequence of random variables in probability

Article

Abstract

In this paper we make a new approach to some well known summability methods using ideals and introducing new notions like \(\mathcal{I }\)-statistical convergence of a sequence of random variables in probability, \(\mathcal{I }\)-lacunary statistical convergence of a sequence of random variables in probability and \(\mathcal{I }\)-\(\lambda \)-statistical convergence of a sequence of random variables in probability. Further we investigate their interrelationship and study some of their important properties.

Keywords

Random variable \(\mathcal{I }\)-statistical convergence \(\mathcal{I }\)-lacunary statistical convergence \(\mathcal{I }\)-\(\lambda \)-statistical convergence \([V, \lambda ]\) (\(\mathcal{I }\))-summability 

Mathematics Subject Classification (2010)

40Axx 40Cxx 60Fxx 60Gxx 

References

  1. 1.
    Bunimovich, L.A.; Sinai, Ya. G.: Statistical properties of lorentz gas with periodic configuration of scatterers, Commun. Math. Phy., Vol-78, No-4, 479–497, doi:10.1007/BF02046760
  2. 2.
    Das, P., Savas, E., Ghosal, S.: On generalization of certain summability methods using ideals. Appl. Math. Lett. 24, 1509–1514 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Erdös, P., Tenenbaum, G.: Sur les densities de certaines suites d’entiers. Proc. London. Math. Soc. 3 59, 417–438 (1989)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MATHMathSciNetGoogle Scholar
  5. 5.
    Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)MATHMathSciNetGoogle Scholar
  6. 6.
    Fridy, J.A., Khan, M.K.: Tauberian theorems via statistical convergence. J. Math. Anal. Appl. 228, 73–95 (1998)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Fridy, J.A., Orhan, C.: Lacunary statistical convergence. Pacific. J. Math. 160, 43–51 (1993)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mountain J. Math. 32, 129–138 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ghosal, S.: Statistical convergence of a sequence of random variables and limit theorems, Accepted for publication in the journal Application of Mathematics (2013). Ref. No. AM 116/2011Google Scholar
  10. 10.
    Ghosal, S.: Lacunary statistical convergence of a sequence of random variables and limit theorems, CommunicatedGoogle Scholar
  11. 11.
    Katetov, M.: Products of filters. Comment. Math. Univ. Carolin 9, 173–189 (1968)MATHMathSciNetGoogle Scholar
  12. 12.
    Kolk, E.: The statistical convergence in Banach spaces. Tartu Ul. Toimetised 928, 41–52 (1991)MathSciNetGoogle Scholar
  13. 13.
    Kostyrko, P., Mačaj, M., Šalát, T., Strauch, O.: On statistical limit points. Proc. Amer. Math. Soc. 129, 2647–2654 (2001)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Leindler, L.: Uber die de la Vallee-Pousnsche Summierbarkeit allge meiner orthogonalreihen. Acta. Math. Acad. Sci. Hungarica 16, 375–387 (1965)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Li, J.: Lacunary statistical cinvergence and inclusion properties between lacunary methods. Int. J. Math. Sci. 23(3), 175–180 (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Maddox, I.J.: Statistical convergence in a locally convex space. Math. Proc. Camb. Phil. Soc. 104, 141–145 (1988)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Martinez, V.G., Torrubia, G.S., Blanc, C.T.: A statistical convergence application for the Hopfield networks. Info. Theo. and Appl 15(1), 84–88 (2008)Google Scholar
  18. 18.
    Miller, H.I.: A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347, 1811–1819 (1995)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Mursaleen, M.: \(\lambda \)-statistical convergence. Math. Slovaca 50, 111–115 (2000)MATHMathSciNetGoogle Scholar
  20. 20.
    Penrose, M.D., Yukich, J.E.: Weak Laws of Large Numbers in Geometric Probability. The Ann. Appl. Prob 13(1), 277–303 (2003)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Pehlivan, S., Mamedov, M.A.: Statistical cluster points and turnpike. Optimization 48, 93–106 (2000)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rohatgi, V. K.: An introduction to probability theory and mathematical statistics, Second ed, Wiley Eastern LimitedGoogle Scholar
  23. 23.
    Šalát, T.: On Statistically convergent sequences of real numbers. Math. Slovaca 30, 139–150 (1980)MATHMathSciNetGoogle Scholar
  24. 24.
    Savas, E.: On statistically convergent sequences of fuzzy numbers. Info. Sci 137, 277–282 (2001)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Savas, E., Das, P.: A generalized statistical convergence via ideals. Appl. Math. Lett. 24, 826–830 (2011)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Schoenberg, I.J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66, 361–375 (1959)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math 2, 73–74 (1951)MathSciNetGoogle Scholar
  28. 28.
    Zygmund, A.: Trigonometric series, 2nd edn. Cambridge Univ. Press, Cambridge (1979)Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKalyani Government Engineering CollegeKalyani, NadiaIndia

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