Rough Küme Teorisinde Tanımlı İki Değişkenli ϖ Dereceden Lacunary İstatistiksel Yakınsama

Abstract

Belirsizlik içeren dizilerde yakınsama yaklaşımı, ilk olarak Pawlak tarafından belirsiz ve muğlak bilgileri ele almak amacıyla formüle edilen rough küme teorisinin gelişimiyle birlikte giderek daha fazla ilgi görmüştür. Bu temele dayanarak, Liu’nun rough dizilerin yakınsamasına ilişkin tanımı da dahil olmak üzere çeşitli genellemeler ortaya konmuştur. Bu çalışmada, rough yakınsama kavramı iki değişkenli dizilere genişletilerek, rough değişkenler için ϖ dereceden lacunary istatistiksel yakınsama adı verilen yeni bir yaklaşım sunulmaktadır. Bu yaklaşım, lacunary dizilerin düzensiz indeks yapısını rough iki değişkenli verilerin yapısıyla birleştirerek daha kapsamlı ve uyarlanabilir bir yakınsama çerçevesi sağlamaktadır. Önerilen yöntem ayrıca klasik toplamlanabilme (summability) teknikleriyle karşılaştırılmış ve karmaşık belirsiz veri yapılarıyla başa çıkmadaki potansiyel uygulamalarını ortaya koyan teorik sonuçlar aracılığıyla analiz edilmiştir.

Keywords

• Lacunary istatistiksel yakınsama
• ϖ dereceden iki değişkenli istatistiksel yakınsama
• rough değişkenler

On the Bivariate Lacunary Statistical Convergence of Order ϖ in Rough Set Theory

Abstract

The approach to convergence for sequences involving uncertainty has gained increasing attention with the development of rough set theory, originally formulated by Pawlak to address imprecise and vague information. Using this foundation as a basis, various generalizations have been introduced, including Liu's definition of the convergence of rough sequences. In this article we extend the notion of rough convergence to the concept of bivariate sequences, introducing a new concept of lacunary statistical convergence of order ϖ for rough variables. This approach integrates the irregular indexing of lacunary sequences with the structure of rough bivariate data, offering a more comprehensive and adaptable convergence framework. The proposed method is also compared with classical summability techniques, and its behavior is analyzed through theoretical results that demonstrate its potential applications in dealing with complex uncertain data structures.

Keywords

• lacunary statistical convergence
• bivariate statistical convergence of order ϖ
• rough variables

References

1. Avşar, B., Savaş, E. 2021. Statistical Convergence of Rough Variable. Konuralp Journal of Mathematics, 9 (2), 268-273.
2. Çetin, S., Kişi, Ö., Gürdal, M. 2025. Exploration of Novel Convergence Concepts for Sequences in Octonion-Valued Metric Spaces. Advances in Mathematical Sciences and Applications, 34 (1), 271-300.
3. Çolak, R. 2010. Statistical Convergence of Order α. Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Publication, 121-129.
4. Das, P., Savas, E. 2014. On I-statistical and I-Lacunary Statistical Convergence of Order α. Bulletin of the Iranian Mathematical Society, 40 (2), 459-472.
5. Das, P., Malik, P. 2008. On Extremal I-Limit Points of Double Sequences. Tatra Mountains Mathematical Publications, (40), 91-102.
6. Fast, H. 1951. Sur la Convergence Statistique. Colloquim Mathematicum, (2), 241-244.
7. Et, M., Cinar, M., Karatas, M. 2013. On-Statistical Convergence of Order α of Sequences of Function. Journal of Inequalities and Applications, (2013), Article ID 204, 1-8.
8. Gadjiev, A. D., Orhan, C. 2002. Some Approximation Theorems via Statistical Convergence. Rocky Mountain Journal of Mathematics, 32, 129-138.

9. Gürdal, M., Huban, M. B. 2014. On I-convergence of Double Sequences in the Topology Induced by Random 2-norms. Mathematicki Vesnik, 66 (1), 73-83.
10. Gürdal, M., Sahiner, A. 2008. Extremal I-Limit Points of Double Sequences. Applied Mathematics E-Notes, (8), 131-137.
11. Gürdal, M., Savas, E. 2022, An Investigation on the Triple Ideal Convergent Sequences in Fuzzy Metric Spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71 (1), 13-24.
12. Huban, M. B., Gürdal, M. 2021. Deferred Invariant Statistical Convergent Triple Sequences via Orlicz Function. Bulletin of Mathematical Analysis and Applications 13 (3), 25-38.
13. Kişi, Ö., Akbıyık, R. 2025. Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order p ̅. Süleyman Demirel University Journal of Natural and Applied Sciences, 29 (1), 35–43.
14. Kişi, Ö., Gürdal, M., Akbıyık, R. 2024. Some Results on Rough I₂-Lacunary Statistical Convergence of Complex Uncertain Sequences. Journal of the Iranian Mathematical Society, 5 (2), 171–189.
15. Kişi, Ö., Çetin, S., Gürdal, M. 2025. Octonion-Valued b-metric Spaces and Ideal Convergence. Journal of Nonlinear Sciences and Applications, 18 (3), Article 02.
16. Kişi, Ö., Çetin, S., & Gürdal, M. 2025. Generalized Statistical Convergence via Modulus Function in Octonion-valued b-metric Spaces. Süleyman Demirel University Journal of Faculty of Engineering and Architecture, 20 (1), 103-125.
17. Liu, B., Liu, Y.K. 2002. Expected Value of Fuzzy Variable and Fuzzy Expected Value Models. IEEE Transactions on Fuzzy Systems, 10 (4), 445-450.
18. Liu, B. 2003. Inequalities and Convergence Concepts of Fuzzy and Rough Variables. Fuzzy Optimization and Decision Making, 2, 87-100.
19. Malik, P., Maity, P. 2013. On Rough Convergence of Double Sequence in Normed Linear Spaces. Afrika Matematika, 28 (1), 89-99.
20. Maity, M. 2014. A note on Rough Statistical Convergence of order α. Journal of Pure Mathematics, 31, 37-46.
21. Mursaleen, M., Cakan, C., Mohiuddine, S. A., Savas. 2010. Generalized Statistical Convergence and Statistical Core of Double Sequences. Acta Mathematica Sinica, English Series, 26, 2131-2144.
22. Mursaleen, M., Edely, O. H. H. 2003. Statistical Convergence of Double Sequences. Journal of Mathematical Analysis and Application, 288, 223-231.
23. Patterson, R. F., Savaş, E. 2000. Lacunary Statistical Convergence of Double Sequences. Mathematical Communication, 10, 55-61.
24. Pawlak, Z. 1982. Rough Sets. International Journal of Computer and Information Sciences, 11 (5), 341-356.
25. Pringsheim, A. 1900. Zur Theorie der Zweifach Unendlichen Zahlen Folgen. Mathematische Annalen, (53), 289-321.
26. Qiu, X. L., Çetin, S., Kişi, Ö., Gürdal, M., Cai, Q.B. 2025. Octonion-valued b-metric Spaces and Results on Its Application. AIMS Mathematics, 10 (5), 10504-10527.
27. Quan, J. J., Çetin, S., Kişi, Ö., Gürdal, M., Cai, Q. B. 2025. On Statistical Convergence in Fractal Analysis. AIMS Mathematics, 10 (8), 18197-18215.
28. Savaş, E. 2016. I_λ-Double Statistical Convergence of order α in Topological Groups. Ukrainian Mathematical Journal, (68), 1251-1258.
29. Savaş, E. 2023. Lacunary Statistical Convergence of Rough Variables in Trust. Journal of Uncertain Systems, 16 (2), 2350003.
30. Savaş, R. 2024. Multidimensional Lacunary Statistical Convergence of Rough Variables in Trust. Proceeding of International Conference on Mathematical Advances and Applications, 1(1), 241-247.
31. Savaş, E., Kişi, Ö., Gürdal, M. 2022. On statistical convergence in Credibility Space. Journal of Intelligent & Fuzzy Systems, 43 (8), 987-1008.
32. Schoenberg, I. J. 1959. The Integrability of Certain Functions and Related Summability Methods. The American Mathematical Monthly, (66), 361-375.
33. Slowinski, R. Vanderpooten, D. A. 2000. Generalized Definition of Rough Approximations Based on Similarity. IEEE Transactions on Knowledge and Data Engineering, 12 (2), 331-336.
34. Şengül, H., Et, M. 2014. On Lacunary Statistical Convergence of Order α^*. Acta Mathematica, 34B (2), 473-482.
35. Yalmancı, U., Gürdal, M. 2015. On Asymptotically Generalized Statistical Equivalent Double Sequences via Ideals. Electronic Journal of Mathematical Analysis and Applications, 3 (1), 89-96.
36. Zadeh, L. A. 1965. Fuzzy Sets. Information and Control, (8), 338-353.
37. Zadeh, L. A. 1978. Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems, 1, 3-28.