A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space


In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set- valued random variables in the sense of the Hausdorff metric dH, based on the result of single-valued random variable obtained by Taylor [1].

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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. doi: 10.4236/jamp.2015.37097.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Taylor, R.L. (1978) Lecture Notes in Mathematics. Springer-Verlag.
[2] Artstein, Z. and Vitale, R.A. (1975) A Strong Law of Large Numbers for Random Compact Sets. Ann. Probab., 3, 879-882. http://dx.doi.org/10.1214/aop/1176996275
[3] Aumann, R. (1965) Integrals of Set Valued Functions. J. Math. Anal. Appl., 12, 1-12. http://dx.doi.org/10.1016/0022-247X(65)90049-1
[4] Hiai, F. and Umegaki, H. (1977) Integrals, Conditional Expectations and Martingales of Multivalued Functions. J. Multivar. Anal., 7, 149-182. http://dx.doi.org/10.1016/0047-259X(77)90037-9
[5] Jung, E.J. and Kim, J.H. (2003) On Set-Valued Stochastic Integrals. Stoch. Anal. Appl., 21, 401-418. http://dx.doi.org/10.1081/SAP-120019292
[6] Li, S., Ogura, Y. and Kreinovich, V. (2002) Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-015-9932-0
[7] Taylor, R.L. and Inoue, H. (1985) A Strong Law of Large Numbers for Random Sets in Banach Spaces. Bull. Instit. Math. Academia Sinica, 13, 403-409.
[8] Hiai, F. (1984) Strong Laws of Large Numbers for Multivalued Random Variables, Multifunctions and Integrands. In: Salinetti, G., Ed., Lecture Notes in Math., Springer, Berlin, 1091, 160-172.
[9] Giné, E., Hahn, G. and Zinn, J. (1983) Limit Theorems for Random Sets: An Application of Probability in Banach Space Results. Lect. Notes in Math., 990, 112-135. http://dx.doi.org/10.1007/bfb0064267
[10] Puri, M.L. and Ralescu, D.A. (1983) Strong Law of Large Numbers for Banach Space Valued Random Sets. Ann. Probab., 11, 222-224. http://dx.doi.org/10.1214/aop/1176993671
[11] Beer, G. (1993) Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications. Kluwer Academic Pub-lishers, Dordrecht, Holland. http://dx.doi.org/10.1007/978-94-015-8149-3
[12] It?, K. and Nisio, M. (1968) On the Convergence of Sums of Independent Banach Space Valued Random Variables. Osaka J. Math., 5, 35-48.

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