Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense

Annals of Probability

Volumn 26, Issue 3, 1998, Pages 1384-1402

  Li, Shoumei ^a   Ogura, Yukio ^a  

^a SAGA UNIVERSITY   (Japan)

Author keywords

Kudo Aumann integral; Kuratowski Mosco convergence; Set valued submartingale; Set valued supermartingale

Indexed keywords

EID: 0032384177     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1022855757     Document Type: Article

References (26)

1. AUMANN, R. J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12 1-12.
2. CHATTERJI, S. D. (1968). Martingale convergence and the RN-theorems. Math. Scand. 22 21-41.
3. DEBREU, G. (1966). Integration of correspondences. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 351-372. Univ. California Press, Berkeley.
4. DUNFORD, N. and SCHWARTZ, J. T. (1958). Linear Operators, Part 1: General Theory. Interscience, New York.
5. HIAI, F. and UMEGAKI, H. (1977). Integrals, conditional expectations and martingales of multivalued functions. J. Multivariate Anal. 7 149-182.
6. HIAI, F. (1978). Radon-Nikodym theorem for set-valued measures. J. Multivariate Anal. 8 96-118.
7. HIAI, F. (1985). Convergence of conditional expectations and strong laws of large numbers for multivalued random variables. Trans. Amer. Math. Soc. 291 613-627.
8. KLEE, V. L. (1951). Convex sets in linear spaces 2. Duke Math. J. 18 875-883.
9. KLEIN, E. and THOMPSON, A. C. (1984). Theory of Correspondences Including Applications to Mathematical Economics. Wiley, New York.
10. KLEMENT, E. P., PURI, M. L., and RALESCU, D. A. (1986). Limit theorems for fuzzy random variables. Pro. Roy. Soc. London Ser. A 407 171-182.
11. KUDO, H. (1953). Dependent experiments and sufficient statistics. Natural Science Report, Ochanomizu University 4 151-163.
12. KURATOWSKI, K. (1965). Topology 1. Academic Press, New York.
13. LI, S. and OGURA, Y. (1996). Fuzzy random variables, conditional expectations and fuzzy martingales. J. Fuzzy Math. 4 905-9217.
14. LI, S. and OGURA, Y. (1997). Convergence of set valued and fuzzy valued martingales. Fuzzy Sets and Systems. To appear.
15. LI, S. and OGURA, Y. (1997). An optional sampling theorem for fuzzy valued martingales. Proceedings of IFSA'97 (Prague) 4 9-14.
16. LUU, D. Q. (1981). Representations and regularity of multivalued martingales. Acta Math. Vietnam 6 29-40.
17. MOSCO, U. (1969). Convergence of convex set and of solutions of variational inequalities. Adv. Math. 3 510-585.
18. MOSCO, U. (1971). On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl. 35 518-535.
19. PAPAGEORGIOU, N. S. (1985). On the theory of Banach space valued multifunctions 1. Integration and conditional expectation. J. Multivariate Anal. 17 185-206.
20. PAPAGEORGIOU, N. S. (1985). On the theory of Banach space valued multifunctions 2. Set valued martingales and set valued measures. J. Multivariate Anal. 17 207-227.
21. PURI, M. L. and RALESCU, D. A. (1986). Fuzzy random variables. J. Math. Anal. Appl. 114 406-422.
22. PURI, M. L. and RALESCU, D. A. (1991). Convergence theorem for fuzzy martingales. J. Math. Anal. Appl. 160 107-121.
23. PURI, M. L. and RALESCU, D. A. (1983). Strong law of large numbers for Banach space valued random sets. Ann. Probab. 11 222-224.
24. SALINETTI, G. and ROGER J. B. WETS (1977). On the relations between two types of convergence for convex functions. J. Math. Anal. Appl. 60 211-226.
25. SALINETTI, G. and ROGER J. B. WETS (1981). On the convergence of closed-valued measurable multifunctions. Trans. Amer. Math. Soc. 226 275-289.
26. THOBIE, C. G. (1974). Selections de multimesures, application à un théorème de Radon-Nikodym multivoque. C. R. Acad. Sci. Paris Ser. A 279 603-606.