×

zbMATH — the first resource for mathematics

A convergence theorem of fuzzy-valued martingales in the extended Hausdorff metric \(\mathbf {H}_{\infty}\). (English) Zbl 1020.60036
A new embedding method for fuzzy-valued random variables with the same expectation is used to prove a convergence theorem for fuzzy-valued martingales in the sense of \(H_\infty\). The considered fuzzy subsets do not need to satisfy the Lipschitz condition.

MSC:
60G48 Generalizations of martingales
03E72 Theory of fuzzy sets, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Artstein, Z.; Vitale, R.A., A strong law of large numbers for random compact sets, Ann. probab., 3, 879-882, (1975) · Zbl 0313.60012
[2] Aumann, R., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301
[3] Beer, G., Topologies on closed and closed convex sets, mathematics and its applications, (1993), Kluwer Academic Publishers Dordrecht, Holland · Zbl 0792.54008
[4] Chatterji, S.D., Martingale convergence and the radon – nikodym theorem in Banach spaces, Math. scand., 22, 21-41, (1968) · Zbl 0175.14503
[5] Debreu, G., Integration of correspondences, (), 351-372 · Zbl 0211.52803
[6] Diamond, P.; Kloden, P., Characterization of compact subsets of fuzzy sets, Fuzzy sets and systems, 29, 341-348, (1989) · Zbl 0661.54011
[7] Hess, C., On multivalued martingales whose values may be unboundedmartingale selectors and mosco convergence, J. multivariate anal., 39, 175-201, (1991) · Zbl 0746.60051
[8] Hiai, F., Strong laws of large numbers for multivalued random variables, (), 160-172
[9] Hiai, F.; Umegaki, H., Integrals, conditional expectations and martingales of multivalued functions, J. multivariate anal., 7, 149-182, (1977) · Zbl 0368.60006
[10] Hörmander, L., Sur LES fonction d’appui des ensembles convexes dans une espace localement convexe, Ark. för mat., 3, 181-186, (1954) · Zbl 0064.10504
[11] Klein, E.; Thompson, A.C., Theory of correspondences including applications to mathematical economics, (1984), Wiley New York · Zbl 0556.28012
[12] Klement, E.P.; Puri, L.M.; Ralescu, D.A., Limit theorems for fuzzy random variables, Proc. roy. soc. London, 407, 171-182, (1986) · Zbl 0605.60038
[13] Li, S.; Ogura, Y., Fuzzy random variables, conditional expectations and fuzzy martingales, J. fuzzy math., 4, 905-927, (1996) · Zbl 0879.60001
[14] Li, S.; Ogura, Y., Convergence of set-valued sub- and super-martingales in the kuratowski – mosco sense, Ann. probab., 26, 3, 1384-1402, (1998) · Zbl 0938.60031
[15] Li, S.; Ogura, Y., Convergence of set-valued and fuzzy-valued martingales, Fuzzy sets and systems, 101, 453-461, (1999) · Zbl 0933.60041
[16] S. Li, Y. Ogura, Convergence in graph for fuzzy-valued martingales and smartingales, in: C. Bertoluzza, A.M. Gil, D.A. Ralescu (Eds.), Statistical Modeling, Analysis, and Management of Fuzzy Data, 2002, pp. 72-89.
[17] Li, S.; Ogura, Y.; Kreinovich, V., Limit theorems and applications of set valued and fuzzy valued random variables, (2002), Kluwer Academic Publishers Dordrecht, to be published · Zbl 1348.60003
[18] Li, S.; Ogura, Y.; Nguyen, H.T., Gaussian processes and martingales for fuzzy-valued variables with continuous parameter, Inform. sci., 133, 7-21, (2001) · Zbl 0988.60025
[19] Ma, M., On embedding problems of fuzzy number spacepart 5, Fuzzy sets and systems, 55, 313-318, (1993)
[20] Ogura, Y.; Li, S., Separability for graph convergence of sequences of fuzzy-valued random variables, Fuzzy sets and systems, 123, 19-27, (2001) · Zbl 0994.60035
[21] Ogura, Y.; Li, S.; Ralescu, D., Set defuzzification and Choquet integral, J. uncertainty, fuzziness and knowledge-based systems, 9, 1-12, (2001) · Zbl 1113.03342
[22] Y. Ogura, S. Li, A strong law of large numbers for fuzzy-valued random variables in the extended Hausdorff metric \( H∞\), preprint.
[23] Papageorgiou, N.S., On the theory of Banach space valued multifunctions, 1, integration and conditional expectation, J. multivariate anal., 17, 185-206, (1985) · Zbl 0579.28009
[24] F.N. Proske, M.L. Puri, Limit theorems for fuzzy random variables, to appear. · Zbl 0990.60020
[25] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 406-422, (1986) · Zbl 0605.60038
[26] Puri, M.L.; Ralescu, D.A., Convergence theorem for fuzzy martingales, J. math. anal. appl., 160, 107-121, (1991) · Zbl 0737.60005
[27] Rȧdström, H., An embedding theorem for spaces of convex sets, Proc. amer. math. soc., 3, 165-169, (1952) · Zbl 0046.33304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.