Geodesic
Further results on laws of large numbers for uncertain random variables
Kybernetika, Tome 59 (2023) no. 2, pp. 314-338
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent and identically distributed random variables and uncertain variables without satisfying the conditions of regular, independent and identically distributed (IID). Two kinds of Marcinkiewicz-Zygmund type LLNs for uncertain random variables were established in the case of $p \in (0, 1)$ by the second theorem, and in the case of $p > 1$ by the third theorem, respectively. For better illustrating of LLNs for uncertain random variables, some examples were stated and explained. Compared with the existed theorems of LLNs for uncertain random variables, our theorems are the generalised results.
The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent and identically distributed random variables and uncertain variables without satisfying the conditions of regular, independent and identically distributed (IID). Two kinds of Marcinkiewicz-Zygmund type LLNs for uncertain random variables were established in the case of $p \in (0, 1)$ by the second theorem, and in the case of $p > 1$ by the third theorem, respectively. For better illustrating of LLNs for uncertain random variables, some examples were stated and explained. Compared with the existed theorems of LLNs for uncertain random variables, our theorems are the generalised results.
DOI : 10.14736/kyb-2023-2-0314
Classification : 46A45, 60F15
Keywords: law of large numbers; uncertain random variable; Etemadi type theorem; Marcinkiewicz–Zygmund type theorem 
@article{10_14736_kyb_2023_2_0314,
     author = {Hu, Feng and Fu, Xiaoting and Qu, Ziyi and Zong, Zhaojun},
     title = {Further results on laws of large numbers for uncertain random variables},
     journal = {Kybernetika},
     pages = {314--338},
     year = {2023},
     volume = {59},
     number = {2},
     doi = {10.14736/kyb-2023-2-0314},
     mrnumber = {4600380},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2023-2-0314/}
}
TY  - JOUR
AU  - Hu, Feng
AU  - Fu, Xiaoting
AU  - Qu, Ziyi
AU  - Zong, Zhaojun
TI  - Further results on laws of large numbers for uncertain random variables
JO  - Kybernetika
PY  - 2023
SP  - 314
EP  - 338
VL  - 59
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2023-2-0314/
DO  - 10.14736/kyb-2023-2-0314
LA  - en
ID  - 10_14736_kyb_2023_2_0314
ER  - 
%0 Journal Article
%A Hu, Feng
%A Fu, Xiaoting
%A Qu, Ziyi
%A Zong, Zhaojun
%T Further results on laws of large numbers for uncertain random variables
%J Kybernetika
%D 2023
%P 314-338
%V 59
%N 2
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2023-2-0314/
%R 10.14736/kyb-2023-2-0314
%G en
%F 10_14736_kyb_2023_2_0314
Hu, Feng; Fu, Xiaoting; Qu, Ziyi; Zong, Zhaojun. Further results on laws of large numbers for uncertain random variables. Kybernetika, Tome 59 (2023) no. 2, pp. 314-338. doi: 10.14736/kyb-2023-2-0314

[1] Ahmadzade, H., Sheng, Y., Esfahani, M.: On the convergence of uncertain random sequences. Fuzzy Optim. Decis. Making 16 (2017), 205-220. | DOI | MR

[2] Avena, L., Hollander, F. den, Redig, F.: Law of large numbers for a class of random walks in dynamic random environments. Electron. J. Probab. 16 (2011), 587-617. | DOI | MR

[3] Bretagnolle, J., Klopotowski, A.: Sur l' existence des suites de variables aléatoires s à s indépendantes échangeables ou stationnaires. Ann. De L'IHP Probab. Et Statist. 31 (1995), 325-350. | MR

[4] Calvo, T., Mesiarová, A., Valášková, L.: Construction of aggregation operators: New composition method. Kybernetika 39 (2003), 643-650. | DOI | MR

[5] Comets, F., Zeitouni, O.: A law of large numbers for random walks in random mixing environments. Ann. Probab. 32 (2004), 880-914. | DOI | MR

[6] Hollander, F. den, Santos, R. dos, Sidoravicius, V.: Law of large numbers for non-elliptic random walks in dynamic random environments. Stoch. Process. Appl. 123 (2013), 156-190. | DOI | MR

[7] Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheorie Und Verwandte Geb. 55 (1981), 119-122. | DOI | MR

[8] Fullér, R.: A law of large numbers for fuzzy numbers. Fuzzy Set. Syst. 45 (1992), 299-303. | DOI | MR

[9] Gao, Y.: Uncertain inference control for balancing inverted pendulum. Fuzzy Optim. Decis. Making 11 (2012), 481-492. | DOI | MR

[10] Gao, J., Yao, K.: Some concepts and theorems of uncertain random process. Int. J. Intell. Syst. 30 (2015), 52-65. | DOI

[11] Gao, R., Ahmadzade, H.: Further results of convergence of uncertain random sequences. Iran. J. Fuzzy Syst. 15 (2018), 31-42. | MR

[12] Gao, R., Sheng, Y.: Law of large numbers for uncertain random variables with different chance distributions. J. Intell. Fuzzy Syst. 31 (2016), 1227-1234. | DOI

[13] Gao, R., Ralescu, D. A.: Convergence in distribution for uncertain random variables. IEEE Trans. Fuzzy Syst. 26 (2018), 1427-1434. | DOI

[14] Hong, D. H., Ro, P. I.: The law of large numbers for fuzzy numbers with unbounded supports. Fuzzy Set. Syst. 116 (2000), 269-274. | DOI | MR

[15] Hou, Y.: Subadditivity of chance measure. J. Uncertain. Anal. Appl. 2 (2014), 14. | DOI

[16] Inoue, H.: A strong law of large numbers for fuzzy random sets. Fuzzy Set. Syst. 41 (1991), 285-291. | DOI | MR

[17] Joo, S. Y., Kim, Y. K.: Kolmogorov's strong law of large numbers for fuzzy random variables. Fuzzy Set. Syst. 120 (2001), 499-503. | DOI | MR

[18] Ke, H., Su, T., Ni, Y.: Uncertain random multilevel programming with application to product control problem. Soft Comput. 19 (2015), 1739-1746. | DOI

[19] Ke, H., Ma, J., Tian, G.: Hybrid multilevel programming with uncertain random parameters. J. Intell. Manuf. 28 (2017), 589-596. | DOI

[20] Kim, Y. K.: A strong law of large numbers for fuzzy random variables. Fuzzy Set. Syst. 111 (2000), 319-323. | DOI | MR

[21] Komorowski, T., Krupa, G.: The law of large numbers for ballistic, multidimensional random walks on random lattices with correlated sites. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 263-285. | MR

[22] Kruse, R.: The strong law of large numbers for fuzzy random variables. Inf. Sci. 28 (1982), 233-241. | DOI | MR

[23] Kwakernaak, H.: Fuzzy random variables-I. definitions and theorems. Inf. Sci. 15 (1978), 1-29. | DOI | MR

[24] Kwakernaak, H.: Fuzzy random variables-II: Algorithms and examples for the discrete case. Inf. Sci. 17 (1979), 253-278. | DOI | MR

[25] Liu, B.: Uncertainty Theory. Second edition. Springer-Verlag, Berlin 2007. | MR

[26] Liu, B.: Fuzzy process, hybrid process and uncertain process. J. Uncertain Syst. 2 (2008), 3-16.

[27] Liu, B.: Some research problems in uncertainty theory. J. Uncertain Syst. 3 (2009), 3-10. | MR

[28] Liu, B.: Theory and Practice of Uncertain Programming. Second edition. Springer-Verlag, Berlin 2009.

[29] Liu, B.: Uncertain set theory and uncertain inference rule with application to uncertain control. J. Uncertain Syst. 4 (2010), 83-98.

[30] Liu, B.: Why is there a need for uncertainty theory?. J. Uncertain Syst. 6 (2012), 3-10.

[31] Liu, B.: Uncertain random graph and uncertain random network. J. Uncertain Syst. 8 (2014), 3-12.

[32] Liu, B., Chen, X.: Uncertain multiobjective programming and uncertain goal programming. J. Uncertainty Anal. Appl. 3 (2015), 10. | DOI

[33] Liu, Y.: Uncertain random variables: A mixture of uncertainty and randomness. Soft Comput. 17 (2013), 625-634. | DOI

[34] Liu, Y.: Uncertain random programming with applications. Fuzzy Optim. Decis. Making 12 (2013), 153-169. | DOI | MR

[35] Liu, Y., Ralescu, D. A.: Risk index in uncertain random risk analysis. Int. J. Uncertain. Fuzz. 22 (2014), 491-504. | DOI | MR

[36] Liu, Y., Yao, K.: Uncertain random logic and uncertain random entailment. J. Ambient Intell. Hum. Comput. 8 (2017), 695-706. | DOI

[37] Loève, M.: Probability Theory I. Second edition. Springer, New York 1977. | DOI | MR

[38] Miyakoshi, M., Shimbo, M.: A strong law of large numbers for fuzzy random variables. Fuzzy Set. Syst. 12 (1984), 133-142. | DOI | MR | Zbl

[39] Nowak, P., Hryniewicz, O.: On some laws of large numbers for uncertain random variables. Symmetry 13 (2021), 2258. | DOI

[40] Qin, Z.: Uncertain random goal programming. Fuzzy Optim. Decis. Making 17 (2018), 375-386. | DOI | MR

[41] Sheng, Y., Shi, G., Qin, Z.: A stronger law of large numbers for uncertain random variables. Soft Comput. 22 (2018), 5655-5662. | DOI

[42] Struk, P.: Extremal fuzzy integrals. Soft Comput. 10 (2006), 502-505. | DOI

[43] Valášková, L., Struk, P.: Preservation of distinguished fuzzy measure classes by distortion. In: Modeling Decisions for Artificial Intelligence (H. Katagiri, T. Hayashida, I. Nishizaki, J. Ishimatsu, K. S. Tree, A. Solution, eds.), Berlin Heidelberg, Springer 2004, pp. 175-182.

[44] Valášková, L., Struk, P.: Classes of fuzzy measures and distortion. Kybernetika 41 (2005), 205-212. | DOI | MR

[45] Rudin, W.: Real and Complex Analysis. Third edition. McGraw-Hill, NewYork 1987. | MR

[46] Yao, K., Chen, X.: A numerical method for solving uncertain differential equations. J. Intell. Fuzzy Syst. 25 (2013), 825-832. | DOI | MR

[47] Yao, K., Gao, J.: Law of large numbers for uncertain random variables. IEEE Trans. Fuzzy Syst. 24 (2016), 615-621. | DOI

[48] Zadeh, L. A.: Fuzzy sets. Inf. Control 8 (1965), 338-353. | DOI | MR | Zbl

[49] Zadeh, L. A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst. 1 (1978), 3-28. | DOI | MR | Zbl

[50] Zhou, J., Wang, F., Wang, K.: Multi-objective optimization in uncertain random environments. Fuzzy Optim. Decis. Making 13 (2014), 397-413. | DOI | MR

Cité par Sources :