Korean J. Math. Vol. 21 No. 4 (2013) pp.383-399
DOI: https://doi.org/10.11568/kjm.2013.21.4.383

Some notes on strong law of large numbers for Banach space valued fuzzy random variables

Main Article Content

Joo-mok Kim
Yun Kyong Kim

Abstract

We establish two types of SLLN for fuzzy random variables. The first result is SLLN for strong-comactly uniformly integrable fuzzy random variables, and the other is the case of that the averages of its expectations converges.


Article Details

References

[1] Z. Artstein and J. C. Hansen, Convexification in limit laws of random sets in Banach spaces Ann. Probab. 13 (1985), 307–309. Google Scholar

[2] Z. Artstein and R. A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975) 879–882. Google Scholar

[3] A. Colubi, J. S. Dominguez-Menchero, M. Lo pez-Dia z, and M. A. Gil, A gener- alized strong law of large numbers, Probab. Theory Related Fields 114 (1999), 401-417. Google Scholar

[4] A. Colubi, J. S. Dominguez-Menchero, M. Lo pez-Dia z, and R. Ko rner, A method to derive strong laws of large numbers for random upper semicontinuous func- tions, Statist. Probab. Lett. 53 (2001), 269-275. Google Scholar

[5] A. Colubi, J. S. Domingeuz-Menchero, M. L opez-Di az and D. Ralescu, A DE [0, 1] representation of random upper semicontinuous functions, Proc. Amer. Math. Soc. 130 (2002), 3237-3242. Google Scholar

[6] G. Debreu, Integration of correspondences, Proc. 5th Berkeley Symp. Math. Statist. Prob. II 2 (1966), 351–372. Google Scholar

[7] Y. Feng, An approach to generalize law of large numbers for fuzzy random vari- ables, Fuzzy Sets and Systems 128 (2002), 237–245. Google Scholar

[8] K. A. Fu and L. X. Zhang, Strong laws of large numbers for arrays of row-wise independent random compact sets and fuzzy random sets, Fuzzy Sets and Systems 159 (2008), 3360–3368. Google Scholar

[9] G. H. Greco and M. P. Moschen, Supremum metric and relatively compact sets of fuzzy sets, Nonlinear Anal. 64 (2006), 1325–1335. Google Scholar

[10] L. Guan and S. Li, Laws of large numbers for weighted sums of fuzzy set valued random variables, Inter. J. Uncertainty Fuzziness Knowl. Based Syst. 12 (2004), 811–825. Google Scholar

[11] H. Inoue, A strong law of large numbers for fuzzy random sets, Fuzzy Sets and Systems 41 (1991), 285–291. Google Scholar

[12] S. Y. Joo, Strong law of large numbers for tight fuzzy random variables, J. Korean Statist. Soc. 31 (2002), 129–140. Google Scholar

[13] S. Y. Joo and Y. K. Kim, Topological properties on the space of fuzzy sets, J. Math. Anal. Appl. 246 (2000), 576–590. Google Scholar

[14] S. Y. Joo, Y. K. Kim and J. S. Kwon, Strong Convergence for weighted sums of fuzzy random sets, Inform. Sci. 176 (2006), 1086–1099. Google Scholar

[15] Y. K. Kim, Strong law of large numbers for random upper-semicontinuous fuzzy sets, Bull. Korean Math. Soc. 39 (2002), 511–526. Google Scholar

[16] Y. K. Kim, Measurability for fuzzy valued functions, Fuzzy Sets and Systems, 129 (2002), 105–109. Google Scholar

[17] E. P. Klement, M. L. Puri and D. A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. Lond. Ser. A 407 (1986), 171–182. Google Scholar

[18] S. Li and Y. Ogura, Strong laws of large numbers for independent fuzzy set valued random variables, Fuzzy Sets and Systems, 157 (2006), 2569–2578. Google Scholar

[19] S. Li, Y. Ogura and V. Kreinovich, Limit theorems and Applications of set valued and fuzzy set valued random variables, Kluwer Academic Publishers, 2002. Google Scholar

[20] I. S. Molchanov, On strong laws of large numbers for random upper- semicontinuous functions, J. Math. Anal. Appl. 235 (1999), 349–355. Google Scholar

[21] F. Proske and M. L. Puri, Strong laws of large numbers for Banach space valued fuzzy random variables, J. Theoret. Probab. 15 (2002), 543–551. Google Scholar

[22] M. L. Puri and D. A. Ralescu, A strong law of large numbers for Banach space valued random sets, Ann. Probab. 11 (1983), 222–224. Google Scholar

[23] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986), 402–422. Google Scholar

[24] R. L. Taylor and H. Inoue, A strong law of large numbers for random sets in Banach space, Bull. Inst. Math. Acad. Sin. 13 (1985), 403–409. Google Scholar

[25] R. L. Taylor and H. Inoue, Laws of large numbers for random sets, Random sets: Theory and Applications, 1985, IMA 97, Springer. New York, 347–366. Google Scholar

[26] R. L. Taylor and H. Inoue, A strong law of large numbers for random sets in Banach spaces, Bull. Inst. Math. Acad. Sin. 13 (1985), 403–409. Google Scholar

[27] T. Uemura, A law of large numbers for random sets, Fuzzy Sets and Systems, 59 (1993), 181–188. Google Scholar