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.
Similar content being viewed by others
References
Balcerzak M, Dems K and Komisarski A, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715–729
Chang S S, Cho Y J and Kang S M, Nonlinear Operator Theory in Probabilistic Metric Spaces (2001) (New York: Nova Science Publishers)
Constantin G and Istrătescu I, Elements of Probabilistic Analysis with Applications (1989) (Bucharest: Editura Academiei)
Dems K, On \( \mathcal{I} \)-Cauchy sequences, Real Analysis Exchange 30 (2004) 123–128
Fast H, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244
Kostyrko P, Šalát T and Wilczynski W, \( \mathcal{I} \)-convergence, Real Analysis Exchange 26 (2000) 669–685
Menger K, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942) 535–537
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
Šalát T, Tripathy B C and Ziman M, On some properties of \( \mathcal{I} \)-convergence, Tatra Mt. Math. Publ. 28 (2004) 279–286
Schweizer B and Sklar A, Statistical metric spaces, Pacific J. Math. 10 (1960) 313–334
Schweizer B, Sklar A and Thorp E, The metrization of statistical metric spaces, Pacific J. Math. 10 (1960) 673–675
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
Schweizer B and Sklar A, Triangle inequalities in a class of statistical metric spaces, J. London Math. Soc. 38 (1963) 401–406
Schweizer B and Sklar A, Probabilistic Metric Spaces (1983) (New York: Elsevier Science Publishing Co.)
Sempi C, Probabilistic metric spaces, in: Encyclopedia of General Topology (eds) K P Hart et al (2003) (Dordrecht: Kluwer) pp. 288–292
Šerstnev AN,On a probabilistic generalization of metric spaces, Kazan. Gos. Univ.Učen. Zap. 124 (1964) 3–11
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
Steinhaus H, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73–74
Şencçimen C and Pehlivan S, Strong statistical convergence in probabilistic metric spaces, Stoch. Anal. Appl. 26 (2008) 651–664
Tardiff R M, Topologies for probabilistic metric spaces, Pacific J. Math. 65 (1976) 233–251
Thorp E, Generalized topologies for statistical metric spaces, Fund. Math. 51 (1962) 9–21
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-009-0028-x