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Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in G<sub>α</sub> Space
College of Applied Sciences, Beijing University of Technology, Beijing, China
College of Applied Sciences, Beijing University of Technology, Beijing, China
- 1 College of Applied Sciences, Beijing University of Technology, Beijing, China
- 2 College of Applied Sciences, Beijing University of Technology, Beijing, China
Advances in Pure Mathematics·Volume 06 (2016)·Pages 583–592·Published 2 August 2016·DOI10.4236/apm.2016.69047
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Abstract
In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d ∞ H . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].
KeywordsLaws of Large NumbersFuzzy Set-Valued Random VariableHausdorff Metric
- Taylor, R.L. (1978) Lecture Notes in Mathematics. Springer-Verlag, 672.
- Li, G. (2015) A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space. Journal of Applied Mathematics and Physics, 3, 797-801. http://dx.doi.org/10.4236/jamp.2015.37097
- Artstein, Z. and Vitale, R.A. (1975) A Strong Law of Large Numbers for Random Compact Sets. Annals of Probability, 3, 879-882. http://dx.doi.org/10.1214/aop/1176996275
- Hiai, F. (1984) Strong Laws of Large Numbers for Multivalued Random Variables, Multifunctions and Integrands. In: Salinetti, G., Ed., Lecture Notes in Mathematics, Vol. 1091, Springer, Berlin, 160-172.
- Taylor, R.L. and Inoue, H. (1985) A Strong Law of Large Numbers for Random Sets in Banach Spaces. Bulletin of the Institute of Mathematics Academia Sinica, 13, 403-409.
- Colubi, A., López-Díaz, M., Domnguez-Menchero, J.S. and Gil, M.A. (1999) A Generalized Strong Law of Large Numbers. Probability Theory and Related Fields, 114, 401-417. http://dx.doi.org/10.1007/s004400050229
- Feng, Y. (2004) Strong Law of Large Numbers for Stationary Sequences of Random Upper Semicontinuous Functions. Stochastic Analysis and Applications, 22, 1067-1083. http://dx.doi.org/10.1081/SAP-120037631
- Molchanov, I. (1999) On Strong Laws of Large Numbers for Random Upper Semicontinuous Functions. Journal of Mathematical Analysis and Applications, 235, 249-355. http://dx.doi.org/10.1006/jmaa.1999.6403
- Puri, M.L. and Ralescu, D.A. (1991) Convergence Theorem for Fuzzy Martingales. Journal of Mathematical Analysis and Applications, 160, 107-121. http://dx.doi.org/10.1016/0022-247X(91)90293-9
- Li, S. and Ogura, Y. (2003) A Convergence Theorem of Fuzzy Valued Martingale in the Extended Hausdorff Metric H ∞ . Fuzzy Sets and Systems, 135, 391-399. http://dx.doi.org/10.1016/S0165-0114(02)00145-8
- Li, S. and Ogura, Y. (2003) Strong Laws of Numbers for Independent Fuzzy Set-Valued Random Variables. Fuzzy Sets and Systems, 157, 2569-2578. http://dx.doi.org/10.1016/j.fss.2003.06.011
- Inoue, H. (1991) A Strong Law of Large Numbers for Fuzzy Random Sets. Fuzzy Sets and Systems, 41, 285-291. http://dx.doi.org/10.1016/0165-0114(91)90132-A
- Guan, L. and Li, S. (2004) Laws of Large Numbers for Weighted Sums of Fuzzy Set-Valued Random Variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, 811-825. http://dx.doi.org/10.1142/S0218488504003223