Skip to main content
Log in

Strong ideal convergence in probabilistic metric spaces

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

In the present paper we introduce the concepts of strongly ideal convergent sequence and strong ideal Cauchy sequence in a probabilistic metric (PM) space endowed with the strong topology, and establish some basic facts. Next, we define the strong ideal limit points and the strong ideal cluster points of a sequence in this space and investigate some properties of these concepts.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Balcerzak M, Dems K and Komisarski A, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715–729

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang S S, Cho Y J and Kang S M, Nonlinear Operator Theory in Probabilistic Metric Spaces (2001) (New York: Nova Science Publishers)

    MATH  Google Scholar 

  3. Constantin G and Istrătescu I, Elements of Probabilistic Analysis with Applications (1989) (Bucharest: Editura Academiei)

    MATH  Google Scholar 

  4. Dems K, On \( \mathcal{I} \)-Cauchy sequences, Real Analysis Exchange 30 (2004) 123–128

    MathSciNet  Google Scholar 

  5. Fast H, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244

    MATH  MathSciNet  Google Scholar 

  6. Kostyrko P, Šalát T and Wilczynski W, \( \mathcal{I} \)-convergence, Real Analysis Exchange 26 (2000) 669–685

    MathSciNet  Google Scholar 

  7. Menger K, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942) 535–537

    Article  MATH  MathSciNet  Google Scholar 

  8. Saadati R and Amini M, D-boundedness and D-compactness in finite dimensional probabilistic normed spaces, Proc. Indian Acad. Sci. (Math. Sci.) 115 (2005) 483–492

    Article  MATH  MathSciNet  Google Scholar 

  9. Šalát T, Tripathy B C and Ziman M, On some properties of \( \mathcal{I} \)-convergence, Tatra Mt. Math. Publ. 28 (2004) 279–286

    MATH  MathSciNet  Google Scholar 

  10. Schweizer B and Sklar A, Statistical metric spaces, Pacific J. Math. 10 (1960) 313–334

    MATH  MathSciNet  Google Scholar 

  11. Schweizer B, Sklar A and Thorp E, The metrization of statistical metric spaces, Pacific J. Math. 10 (1960) 673–675

    MATH  MathSciNet  Google Scholar 

  12. Schweizer B and Sklar A, Statistical metric spaces arising from sets of random variables in Euclidean n spaces, Theory Prob. Appl. 7 (1962) 447–456

    Article  MATH  MathSciNet  Google Scholar 

  13. Schweizer B and Sklar A, Triangle inequalities in a class of statistical metric spaces, J. London Math. Soc. 38 (1963) 401–406

    Article  MATH  MathSciNet  Google Scholar 

  14. Schweizer B and Sklar A, Probabilistic Metric Spaces (1983) (New York: Elsevier Science Publishing Co.)

    MATH  Google Scholar 

  15. Sempi C, Probabilistic metric spaces, in: Encyclopedia of General Topology (eds) K P Hart et al (2003) (Dordrecht: Kluwer) pp. 288–292

    Chapter  Google Scholar 

  16. Šerstnev AN,On a probabilistic generalization of metric spaces, Kazan. Gos. Univ.Učen. Zap. 124 (1964) 3–11

    Google Scholar 

  17. Sleziak M, Toma V, Čincura J and Šalát T, Sets of statistical cluster points and \( \mathcal{I} \)-cluster points, Real Analysis Exchange 30 (2004) 565–580

    MathSciNet  Google Scholar 

  18. Steinhaus H, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73–74

    MathSciNet  Google Scholar 

  19. Şencçimen C and Pehlivan S, Strong statistical convergence in probabilistic metric spaces, Stoch. Anal. Appl. 26 (2008) 651–664

    Article  MathSciNet  Google Scholar 

  20. Tardiff R M, Topologies for probabilistic metric spaces, Pacific J. Math. 65 (1976) 233–251

    MATH  MathSciNet  Google Scholar 

  21. Thorp E, Generalized topologies for statistical metric spaces, Fund. Math. 51 (1962) 9–21

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Celaleddin Şençimen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Şençimen, C., Pehlivan, S. Strong ideal convergence in probabilistic metric spaces. Proc Math Sci 119, 401–410 (2009). https://doi.org/10.1007/s12044-009-0028-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12044-009-0028-x

Keywords

Navigation