$S_{\lambda }\left( \mathcal{I}\right) $-convergence of
	complex uncertain sequence

O. Kisi

doi:10.15330/ms.51.1.183-194

This study introduces the $\lambda _{\mathcal{I}}$-statistically convergence concepts of complex uncertain sequences: $\lambda _{\mathcal{I}}$% -statistically convergence almost surely ($S_{\lambda }( \mathcal{I}% ) .a.s.$), $\lambda _{\mathcal{I}}$-statistically convergence in measure, $\lambda _{\mathcal{I}}$-statistically convergence in mean, $% \lambda _{\mathcal{I}}$-statistically convergence in distribution and $% \lambda _{\mathcal{I}}$-statistically convergence uniformly almost surely ($% S_{\lambda }( \mathcal{I}) .u.a.s.$). In addition, decomposition theorems and relationships among them are discussed.

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

2. Schoenberg I.J., The integrability of certain functions and related summability methods, Am. Math. Mon., 66 (1959), 361–375.

3. Freedman A.R., Sember J.J., Densities and summability, Pac. J. Math., 95 (1981), №2, 293–305.

4. Chen X., Ning Y., Wang X., Convergence of complex uncertain sequences, Journal of Intelligent & Fuzzy Systems, 30 (2016), №6, 3357–3366.

5. Chen X., American option pricing formula for uncertain financial market, International Journal of Operations Research, 8 (2011), №2, 32–37.

6. Das P., Savas E., Ghosal S. Kr., On generalized of certain summability methods using ideals, Appl. Math. Letter, 36 (2011), 1509-1514.

7. Demirci K., I-limit superior and limit inferior, Math. Commun., 6 (2001), №2, 165-172.

8. Gao X., Gao Y., Connectedness index of uncertain graphs, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, 21 (2013), №1, 127-137.

9. Guo H., Xu C., A necessary and sufficient condition of convergence in mean square for uncertain sequence, Information: An International Interdisciplinary Journal, 16 (2013), №2(A), 1091-1096.

10. Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.

11. Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer- Verlag, Berlin, 2011.

12. Liu B., Theory and Practice of Uncertain Programming, 2nd edn., Springer-Verlag, Berlin, 2009.

13. Liu B., Chen X., Uncertain multiobjective programming and uncertain goal programming, Journal of Uncertainty Analysis and Applications, 3 (2015), Article 10.

14. Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), №1, 3-10.

15. Liu B., Uncertainty risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, 4 (2010), №3, 163-170.

16. Liu B., Uncertainty logic for modelling human language, Journal of Uncertain Systems, 5 (2011), №1, 3-20.

17. Liu B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), №1, 3-16.

18. Liu B., Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1.

19. Yao K., Chen X., A numberical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), №3, 825-832.

20. Zhang B., Peng J., Euler index in uncertain graph, Applied Mathematics and Computation, 218 (2012), №20, 10279-10288.

21. Liu B., Why is there a need for uncertainty theory?, Journal of Uncertain Systems, 6 (2012), №1, 3-10.

22. Liu B., Uncertainity Theory, 4th edn., Springer-Verlag, Berlin, 2015.

23. Salat T., Tripathy B.C., Ziman M., On some properties of I-convergence, Tatra Mt. Math. Publ., 28 (2004), 279-286.

24. Salat T., Tripathy B.C., Ziman M., I-Convergence Field, Tatra Mt. Math. Publ., 28 (2005), 279-286.

25. Liu B., Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), №1.

26. Kostyrko P., Macaj M., Salat T., Sleziak M., I-convergence and extremal I-limit points, Math. Slovaca, 55 (2005), 443-464.

27. Kisi O., Guler E., $\lambda $-Statistically Convergence of Complex Uncertain Sequence, CMES 2019, Antalya, Turkey.

28. Liu B., Theory and Practice of Uncertain Programming, 2nd edn., Springer-Verlag, Berlin, 2009.

29. Mursaleen M., $\lambda $-statistical convergence, Math. Slovaca, 50 (2000), №1, 111-115.

30. Peng Z.X., Complex uncertain variable, Doctoral Dissertation, Tsinghua University, 2012.

31. You C.L., On the convergence of uncertain sequences, Mathematical and Computer Modelling, 49 (2009), 482.487.

32. Tripathy B.C., Nath P.K., Statistical convergence of complex uncertain sequences, New Mathematics and Natural Computation, 13 (2017), №3, 359-374.

33. Zhang Z., Some discussions on uncertain measure, Fuzzy Optimization and Decision Making, 10 (2011), №1, 31-43.

	$S_{\lambda }\left( \mathcal{I}\right) $-convergence of complex uncertain sequence
Author	O. Kisi okisi@bartin.edu.tr Deparment of Mathematics, Bartэn University, Bartin, Turkey
Abstract	This study introduces the $\lambda _{\mathcal{I}}$-statistically convergence concepts of complex uncertain sequences: $\lambda _{\mathcal{I}}$% -statistically convergence almost surely ($S_{\lambda }( \mathcal{I}% ) .a.s.$), $\lambda _{\mathcal{I}}$-statistically convergence in measure, $\lambda _{\mathcal{I}}$-statistically convergence in mean, $% \lambda _{\mathcal{I}}$-statistically convergence in distribution and $% \lambda _{\mathcal{I}}$-statistically convergence uniformly almost surely ($% S_{\lambda }( \mathcal{I}) .u.a.s.$). In addition, decomposition theorems and relationships among them are discussed.
Keywords	$\lambda $-convergence; uncertainty theory; complex uncertain variable; ideal convergence
DOI	doi:10.15330/ms.51.2.183-194
Reference	1. Fast H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. 2. Schoenberg I.J., The integrability of certain functions and related summability methods, Am. Math. Mon., 66 (1959), 361–375. 3. Freedman A.R., Sember J.J., Densities and summability, Pac. J. Math., 95 (1981), №2, 293–305. 4. Chen X., Ning Y., Wang X., Convergence of complex uncertain sequences, Journal of Intelligent & Fuzzy Systems, 30 (2016), №6, 3357–3366. 5. Chen X., American option pricing formula for uncertain financial market, International Journal of Operations Research, 8 (2011), №2, 32–37. 6. Das P., Savas E., Ghosal S. Kr., On generalized of certain summability methods using ideals, Appl. Math. Letter, 36 (2011), 1509-1514. 7. Demirci K., I-limit superior and limit inferior, Math. Commun., 6 (2001), №2, 165-172. 8. Gao X., Gao Y., Connectedness index of uncertain graphs, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, 21 (2013), №1, 127-137. 9. Guo H., Xu C., A necessary and sufficient condition of convergence in mean square for uncertain sequence, Information: An International Interdisciplinary Journal, 16 (2013), №2(A), 1091-1096. 10. Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007. 11. Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer- Verlag, Berlin, 2011. 12. Liu B., Theory and Practice of Uncertain Programming, 2nd edn., Springer-Verlag, Berlin, 2009. 13. Liu B., Chen X., Uncertain multiobjective programming and uncertain goal programming, Journal of Uncertainty Analysis and Applications, 3 (2015), Article 10. 14. Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), №1, 3-10. 15. Liu B., Uncertainty risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, 4 (2010), №3, 163-170. 16. Liu B., Uncertainty logic for modelling human language, Journal of Uncertain Systems, 5 (2011), №1, 3-20. 17. Liu B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), №1, 3-16. 18. Liu B., Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), Article 1. 19. Yao K., Chen X., A numberical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25 (2013), №3, 825-832. 20. Zhang B., Peng J., Euler index in uncertain graph, Applied Mathematics and Computation, 218 (2012), №20, 10279-10288. 21. Liu B., Why is there a need for uncertainty theory?, Journal of Uncertain Systems, 6 (2012), №1, 3-10. 22. Liu B., Uncertainity Theory, 4th edn., Springer-Verlag, Berlin, 2015. 23. Salat T., Tripathy B.C., Ziman M., On some properties of I-convergence, Tatra Mt. Math. Publ., 28 (2004), 279-286. 24. Salat T., Tripathy B.C., Ziman M., I-Convergence Field, Tatra Mt. Math. Publ., 28 (2005), 279-286. 25. Liu B., Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1 (2013), №1. 26. Kostyrko P., Macaj M., Salat T., Sleziak M., I-convergence and extremal I-limit points, Math. Slovaca, 55 (2005), 443-464. 27. Kisi O., Guler E., $\lambda $-Statistically Convergence of Complex Uncertain Sequence, CMES 2019, Antalya, Turkey. 28. Liu B., Theory and Practice of Uncertain Programming, 2nd edn., Springer-Verlag, Berlin, 2009. 29. Mursaleen M., $\lambda $-statistical convergence, Math. Slovaca, 50 (2000), №1, 111-115. 30. Peng Z.X., Complex uncertain variable, Doctoral Dissertation, Tsinghua University, 2012. 31. You C.L., On the convergence of uncertain sequences, Mathematical and Computer Modelling, 49 (2009), 482.487. 32. Tripathy B.C., Nath P.K., Statistical convergence of complex uncertain sequences, New Mathematics and Natural Computation, 13 (2017), №3, 359-374. 33. Zhang Z., Some discussions on uncertain measure, Fuzzy Optimization and Decision Making, 10 (2011), №1, 31-43.
Pages	183-194
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

$S_{\lambda }\left( \mathcal{I}\right) $-convergence of complex uncertain sequence