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Convergence theorem for fuzzy martingales. (English) Zbl 0737.60005
Fuzzy-valued measures and their relationship to fuzzy random variables are studied. The concept of conditional expectation for fuzzy random variable is proposed. The authors prove the Radon-Nikodym theorem in this setting and a convergence theorem for fuzzy martingale.

MSC:
60A99 Foundations of probability theory
60G48 Generalizations of martingales
03E72 Theory of fuzzy sets, etc.
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[1] Artstein, Z, Set-valued measure, Trans. amer. math. soc., 165, 103-125, (1972) · Zbl 0237.28008
[2] Artstein, Z; Hansen, J.C, Convexification in limit laws of random sets in Banach spaces, Ann. probab., 13, 307-309, (1985) · Zbl 0554.60022
[3] Artstein, Z; Vitale, R.A, A strong law of large numbers for random compact sets, Ann. probab., 3, 879-882, (1975) · Zbl 0313.60012
[4] Aumann, R.J, Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301
[5] Debreu, G, Integration of correspondences, (), 351-372 · Zbl 0211.52803
[6] Debreu, G; Schmeidler, D, The Radon-Nikodym derivative of a correspondence, (), 41-56
[7] De Robertis, L; Hartigan, J.A, Bayesian inference using intervals of measures, Ann. statist., 9, 235-244, (1981) · Zbl 0468.62004
[8] Giné, E; Hahn, M.G; Zinn, J, Limit theorems for random sets: an application of probability in Banach space results, (), 112-135
[9] Hiai, F, Radon-Nikodym theorems for set-valued measures, J. multivariate anal., 8, 96-118, (1978) · Zbl 0384.28006
[10] Hiai, F; Umegaki, H, Integrals, conditional expectations, and martingales of multivalued functions, J. multivariate anal., 7, 149-182, (1977) · Zbl 0368.60006
[11] Kendall, D.G, Foundations of a theory of random sets, () · Zbl 0275.60068
[12] Klement, E.P; Puri, M.L; Ralescu, D.A, Limit theorems for fuzzy random variables, (), 171-182 · Zbl 0605.60038
[13] Kruse, R, The strong law of large numbers for fuzzy random variables, Inform. sci., 28, 233-241, (1982) · Zbl 0571.60039
[14] Matheron, G, Randon sets and integral geometry, (1975), Wiley New York · Zbl 0321.60009
[15] Negoita, C.V; Ralescu, D.A, Application of fuzzy sets to systems analysis, (1975), Wiley New York · Zbl 0326.94002
[16] Puri, M.L; Ralescu, D.A, Strong law of large numbers for Banach space valued random sets, Ann. probab., 11, 222-224, (1983) · Zbl 0508.60021
[17] Puri, M.L; Ralescu, D.A, Strong law of large numbers with respect to a set-valued probability measure, Ann. probab., 11, 1051-1054, (1983) · Zbl 0518.62033
[18] Puri, M.L; Ralescu, D.A, Limit theorems for random compact sets in Banach space, (), 151-158 · Zbl 0559.60007
[19] Puri, M.L; Ralescu, D.A, The concept of normality for fuzzy random variables, Ann. probab., 13, 1373-1379, (1985) · Zbl 0583.60011
[20] Puri, M.L; Ralescu, D.A, Fuzzy random variables, J. math. anal. appl., 114, 402-422, (1986) · Zbl 0605.60038
[21] Rieffel, M.A, The Radon-Nikodym theorem for the Bochner integral, Trans. amer. math. soc., 131, 466-487, (1968) · Zbl 0169.46803
[22] Sugeno, M, Fuzzy measures and fuzzy integrals—A survey, (), 89-102
[23] Weil, W, An application of the central limit theorem for Banach space-valued random variables to the theory of random sets, Z. wahrsch. verw. gebiete, 60, 203-208, (1982) · Zbl 0481.60018
[24] Zadeh, L.A, Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606
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