FURTHER RESULTS OF CONVERGENCE OF UNCERTAIN RANDOM SEQUENCES

Document Type : Research Paper

Authors

1 School of Economics and Management, Hebei University of Technology, Tianjin 300401, China

2 Department of Statistics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

Convergence is an issue being widely concerned about. Thus, in this paper, we mainly put forward two types of concepts of convergence in mean and convergence in distribution for the sequence of uncertain random variables. Then some of theorems are proved to show the relations among the three convergence concepts that are convergence in mean, convergence in measure and convergence in distribution. Furthermore, several examples are given to illustrate how we use the theorems to make sure the uncertain random sequence being convergent. Finally, several counterexamples are taken to explain the relations between these different types of convergence.

Keywords


[1] H. Ahmadzade, Y.H. Sheng, M. Esfahani, On the convergence of uncertain random sequences,
Fuzzy Optimitization and Decision Making, 16(2) (2017), 205-220.
[2] H. Ahmadzade, Y. H. Sheng and F. Hassantabar, Some results of moments of uncertain
random variables, Iranian Journal of Fuzzy Systems, 14(2) (2017), 1{21.
[3] X. W. Cheng and Y. K. Liu, The convergence of fuzzy random variables, Proceedings of the
Eleventh International Fuzzy Systems Association World Congress, 1 (2005), 1{8.
[4] R. Gao and Y. H. Sheng, Law of large numbers for uncertain random variables with different
chance distributions, Journal of Intelligent and Fuzzy Systems, 31 (2016), 1227{1234.
[5] R. Gao, Y. Sun and D. A. Ralescu, Order statistics of uncertain random variables with
application to k-out-of-n system, Fuzzy Optimization and Decision Making, 16(2) (2017),
159-181.
[6] R. Gao and K. Yao, Importance index of components in uncertain random systems,
Knowledge-Based Systems, 109 (2016), 208{217.
[7] H. Y. Guo and X. S. Wang, Variance of uncertain random variables, Journal of Uncertainty
Analysis and Applications, 2(6) (2014), 1{7.
[8] D. Kahneman and A. Tversky, Prospect theory: an analysis of decision under risk, Econo-
metrica, 47(2) (1979), 263{292.
[9] H. Kwakernaak, Fuzzy random variables-I: defi nitions and theorems, Information Sciences,
15(1) (1978), 1{29.
[10] H. Kwakernaak, Fuzzy random variables-II: algorithms and examples for the discrete case,
Information Sciences, 17(3) (1979), 253{278.
[11] B. Liu, Random fuzzy dependent-chance programming and its hybrid intelligent algorithm,
Information Sciences, 141(3) (2002), 259{271.
[12] B. Liu, Uncertainty Theory: an Introduction to Its Axiomatic Foundations, Springer, Berlin,
2004.
[13] B. Liu, Uncertainty Theory (2nd ed.), Springer, Berlin, 2007.
[14] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3(1)
(2009), 3-10.
[15] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,
Springer, Berlin, 2010.
[16] B. Liu, Uncertainty Theory (4th ed.), Springer, Berlin, 2015.
[17] Y. H. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft
Computing, 17(4) (2013), 625{634.
[18] Y. H. Liu, Uncertain random programming with applications, Fuzzy Optimization and Deci-
sion Making, 12(2) (2013), 153-169.
[19] Y. H. Liu and M. H. Ha, Expected value of function of uncertain variables, Journal of Un-
certain Systems, 4(3) (2010), 181-186.
[20] Y. K. Liu, Z. Q. Liu and J. W. Gao, The modes of convergence in the approximation of fuzzy
random optimization problems, Soft Computing, 13(2) (2009), 117-125.
[21] Y. H. Sheng and K. Yao, Some formulas of variance of uncertain random variable, Journal
of Uncertainty Analysis and Applications, Article 12, 2 (2014).
[22] K. Yao, A formula to calculate the variance of uncertain variable, Soft Computing, 19(10)
(2014), 2947{2953.
[23] K. Yao and J. W. Gao, Law of large numbers for uncertain random variables, IEEE Trans-
actions on Fuzzy Systems, 24(3) (2016), 615{621.
[24] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
[25] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1
(1978), 3{28