Central limit theorems for generalized set-valued random variables

Journal of Mathematical Analysis and Applications

Volumn 285, Issue 1, 2003, Pages 250-263

  Li, Shoumei ^a   Ogura, Yukio ^b   Proske, Frank N ^c   Puri, Madan L ^d  

^a BEIJING UNIVERSITY OF TECHNOLOGY   (China)

^b SAGA UNIVERSITY   (Japan)

^c UNIVERSITY OF OSLO   (Norway)

^d INDIANA UNIVERSITY   (United States)

Author keywords

Central limit theorem; Empirical processes; Extended Hausdorff metric; Generalized set valued random variable

Indexed keywords

EID: 0141850347     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-247X(03)00397-4     Document Type: Article

References (33)

1. K.L. Arrow, F.H. Hahn, General Competitive Analysis, Holden-Day, 1971.
2. R. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12.
3. G. Beer, Topologies on Closed and Closed Convex Sets, in: Mathematics and Its Applications, Vol. 268, Kluwer Academic, 1993.
4. L. Breiman, Probability, Addison-Wesley, 1968.
5. N. Cressie, A central limit theorem for random sets, Z. Wahrscheinlichkeitsth. Verw. Gebiete 49 (1979) 37-47.
6. Y. Chen, Fuzzy Systems and Mathematics, Huazhong Institute Press of Science and Technology, 1984 (in Chinese).
7. G. Debreu, Integration of correspondences, in: L. Lecam, J. Neyman, E.L. Scott (Eds.), Proc. 5th Berkeley Symposium Math. Statist. and Probability II, Part I, Univ. of California Press, Berkeley, 1966, pp. 351-372.
8. E. Giné, G. Hahn, J. Zinn, Limit theorems for random sets: an application of probability in Banach space results, in: Lecture Notes in Math., Vol. 990, Springer, 1983, pp. 112-135.
9. C. Hess, On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence, J. Multivariate Anal. 39 (1991) 175-201.
10. C. Hess, Multivalued strong laws of large numbers in the slice topology - application to integrands, Set-Valued Anal. 2 (1994) 183-205.
11. F. Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985) 613-627.
12. F. Hiai, H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149-182.
13. L. Hörmander, Sur les fonction d'appui des ensembles convexes dans une espace localement convexe, Ark. Mat. 3 (1954) 181-186.
14. N.C. Jain, M.B. Marcus, Central limit theorems for C(S)-valued random variables, J. Funct. Anal. 19 (1975) 216-231.
15. D.G. Kendall, Foundations of a theory of random sets, in: Stochastic Geometry, Wiley, 1973.
16. E. Klein, A.C. Thompson, Theory of Correspondences, Including Applications to Mathematical Economics, Wiley, 1984.
17. E.P. Klement, L.M. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London 407 (1986) 171-182.
18. S. Li, Y. Ogura, Fuzzy random variables, conditional expectations and fuzzy martingales, J. Fuzzy Math. 4 (1996) 905-927.
19. S. Li, Y. Ogura, Convergence of set valued sub- and super-martingales in the Kuratowski-Mosco sense, Ann. Probab. 26 (1998) 1384-1402.
20. S. Li, Y. Ogura, H.T. Nguyen, Gaussian processes and martingales for fuzzy valued variables with continuous parameter, Inform. Sci. 133 (2001) 7-21.
21. S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic, 2002.
22. N.N. Lyashenko, Limit theorems for sums of independent compact random subsets of an Euclidean space (Russian, English summary), Investigations in the theory of probability distributions, IV, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979) 113-128, 240, 246.
23. U. Mosco, Convergence of convex set and of solutions of variational inequalities, Adv. Math. 3 (1969) 510-585.
24. ∞, preprint, 2002.
25. F.N. Proske, M.L. Puri, Limit theorems for fuzzy random variables, Stochastic Process. Appl. (2002), to appear.
26. M.L. Puri, D.A. Ralescu, Limit theorems for random compact sets in Banach space, Math. Proc. Cambridge Philos. Soc. 97 (1985) 151-158.
27. M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 406-422.
28. R.T. Rockafellar, R. Wets, Variational Analysis, Springer, 1998.
29. A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes, Springer, 1996.
30. W. Weil, An application of the central limit theorem for Banach space valued random variables to the theory of random sets, Z. Wahrscheinlichkeitsth. Verw. Gebiete 60 (1982) 203-208.
31. R.A. Wijsman, Convergence of sequences of convex set, cones and function, Bull. Amer. Math. Soc. 70 (1964) 186-188.
32. R.A. Wijsman, Convergence of sequences of convex sets, cones and functions, part 2, Trans. Amer. Math. Soc. 123 (1966) 32-45.
33. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.