Generalized Deferred Statistical Convergence in Credibility Spaces: New Perspectives in the Ideal Setting

Year 2026, Volume: 14 Issue: 1 , 99 - 107 , 30.04.2026

Ömer Kişi , Rabia Savas

https://izlik.org/JA23AA49NT

Abstract

In this paper, the concepts of I-deferred statistical convergence in credibility of fuzzy variable sequences are defined, and some inclusion relations between I-deferred statistical convergence in credibility and strongly I-deferred Cesàro summability in credibility of fuzzy variable sequences are explored. We observe that in credibility spaces, a sequence is strongly I-deferred Cesàro summable in credibility if and only if the sequence is I-deferred statistically convergent in credibility. Additionally, we introduce the new concept of I-deferred statistically Cauchy sequences in credibility.

Keywords

Credibility space , statistical convergence , ideal convergence

References

• [1] R. P. Agnew, On deferred Ces`aro means, Ann. Math., 3(3) (1932), 413-421.
• [2] Z. Afacan and E. D¨undar, Ideal convergence in 2-metric spaces, Konuralp J. Math., 13(2) (2025), 259-263.
• [3] S. Aydın and H. Polat, Difference sequence spaces derived by using Pascal transform, Fundam. J. Math. Appl., 2(1) (2019), 56-62.
• [4] P. Das, E. Savas¸ and S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett., 24 (2011), 1509-1514.
• [5] E. G¨ulle and U. Ulusu, Wijsman deferred invariant statistical and strong p-deferred invariant equivalence of order a, Fundam. J. Math. Appl., 6(4) (2023), 211-217.
• [6] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
• [7] H. S¸ . Kandemir, M. Et, N. D. Aral, Strongly l-convergence of order a in neutrosophic normed spaces, Dera Natung Govt. College Res. J., 7(1) (2022), 92-102.
• [8] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, 1st Edn., Academic Press, New York, 1975, pp.1-416.
• [9] O¨ . Kis¸i, On I-convergence in neutrosophic normed spaces, Fundam. J. Math. Appl., 4(2) (2021), 67-76.
• [10] O¨ . Kis¸i and E. Gu¨ler, I-Cesa`ro summability of a sequence of order a of random variables in probability, Fundam. J. Math. Appl., 1(2) (2018), 157-161.
• [11] O¨ . Kis¸i and M. Gu¨rdal, Certain aspects of deferred statistical convergence of fuzzy variables in credibility space, J. Anal., 32 (2024), 2057-2075.
• [12] O¨ . Kis¸i, M. Gu¨rdal and E. Savas¸, On deferred statistical convergence of fuzzy variables, Appl. Appl. Math., 17 (2022), 366-385.
• [13] P. Kostyrko, T. ˘ Sal´at and W. Wilczynsski, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
• [14] M. K¨uc¸ ¨ukaslan, M. Yilmazt¨urk, On deferred statistical convergence of sequences, Kyungpook Math. J., 56(2) (2016), 357-366.
• [15] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Trans. Fuzzy Syst., 10 (2002), 445-450.
• [16] X. Li and B. Liu, A sufficient and necessary condition for credibility measures, Int. J. Uncertainty Fuzziness Knowledge-Based Syst., 2006, 14, 527-535.
• [17] X. Li and B. Liu, Chance measure for hybrid events with fuzziness and randomness, Soft Comput., 2008, 13, 105-115.
• [18] B. Liu, Uncertainty Theory, 2nd Edn., Springer, Berlin, 2007, pp.1-246.
• [19] B. Liu, A survey of credibility theory, Fuzzy Optim. Decis. Making, 5 (2006), 387-408.
• [20] B. Liu, Inequalities and convergence concepts of fuzzy and rough variables, Fuzzy Optim. Decis. Making, 2 (2003), 87-100.
• [21] Q. Jiang, Some remarks on convergence in credibility distribution of fuzzy variable, International Conference on Intelligence Science and Information Engineering, 2011, Wuhan, China, pp.446-449.
• [22] S. Ma, The convergence properties of the credibility distribution sequence of fuzzy variables, J. Modern Math. Frontier, 3 (2014), 24-27.
• [23] S. Nahmias, Fuzzy variables, Fuzzy Sets Syst., 1 (1978), 97-110.
• [24] ˙I. Osmano˘glu and E. D¨undar, A note on statistical continuity of functions, Fundam. J. Math. Appl., 7(4) (2024), 212-217.
• [25] E. Savas¸, O¨ . Kis¸i and M. Gu¨rdal, On some generalized deferred statistical convergence of order ab for fuzzy variable sequences in credibility space, Facta Univ., Ser. Math. Inform., 38 (2023), 599-620.
• [26] E. Savas¸ and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24 (2011), 826-830.
• [27] E. Savas¸, O¨ . Kis¸i and M. Gu¨rdal, On statistical convergence in credibility space, Numer. Funct. Anal. Optim., 43 (2022), 987-1008.
• [28] R. Savas¸, Multidimensional statistical convergence in credibility theory, Int. J. Uncertainty Fuzziness Knowledge-Based Syst., 31(4) (2023) 571-583.
• [29] R. Savas¸ and R. F. Patterson, Multidimensional matrix characterization of asymptotic I2-equivalent and ideal for double sequences, Numer. Funct. Anal. Optim., 40(6) (2019), 654-669.
• [30] R. Savas¸ and M. O¨ ztu¨rk, On generalized ideal asymptotically statistical equivalent of order a for functions, Ukr. Math. J., 70(12) (2018), 1650-1659.
• [31] H. S¸eng¨ul, M. Et, and M. Is¸ık, On I-deferred statistical convergence, Conference Proceedings of ICMS-18, Maltepe/Istanbul (2019).
• [32] U. Ulusu, F. Nuray, and E. D¨undar, I-limit and I-cluster points for functions defined on amenable semigroups, Fundam. J. Math. Appl., 4(1) (2021), 45-48.
• [33] T. Yaying, On L-Fibonacci difference sequence spaces of fractional order, Dera Natung Govt. College Res. J. 6(1) (2021), 92–102.
• [34] T. Yaying, Arithmetic continuity in cone metric space, Dera Natung Govt. College Res. J., 5(1) (2020), 55–62.
• [35] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338-353.