Abstract
This paper tries to give a systematic investigation of integration of random fuzzy sets. Besides the widely used Aumann-integral adaptions of Pettis- and Bochner-integration for random elements in Banach spaces are introduced. The mutual relationships of these competing concepts will be explored comprehensively, completing and improving former results from literature. As a by product dominated convergence theorems, strong laws of large numbers and central limit theorems for random fuzzy sets can be derived. They are based on weaker assumptions than previous versions from literature.
Similar content being viewed by others
References
Aumann, R. J. (1965). Integrals of set-valued functions.Journal of Mathematical Analysis and Applications, 12:1–12.
Cohn, D. L. (1997).Measure Theory. Birkhäuser, Boston.
Colubi, A., López-Díaz, M., Domínguez-Menchero, J. S., andGil, M. A. (1999). A generalized strong law of large numbers.Probability Theory and Related Fields, 114:401–417.
Debreu, G. (1967). Integration of correspondences. InProceedings of the Fifth Berkely Symposium on Mathematical Statistics and Probability, Vol. II, Part I, pp. 351–372. University of California Press, Berkeley/Los Angeles.
Diamond, P. andKloeden, P. (1990). Metric spaces of fuzzy sets.Fuzzy Sets and Systems,35:241–249.
Diamond, P. andKloeden, P. (1994).Metric Spaces of Fuzzy Sets. World Scientific, Singapore.
Engelking, R. (1989).General Topology. Heldermann, Berlin.
Etemadi, N. (1981). An elementary proof of the strong law of large numbers.Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 55:119–122.
Hiai, F. andUmegaki, H. (1977). Integrals, conditional expectations, and martingales of multivalued functions.Journal of Multivariate Analysis, 7:149–182.
Hoffmann-Jørgensen, J. (1985). The law of large numbers for nonmeasurable and non-separable random elements.Astérisque, 131:299–356.
Hoffmann-Jørgensen, J. andPisier, G. (1976). The law of large numbers and the central limit theorem in Banach spaces.The Annals of Probability, 4:587–599.
Klein, E. andThompson, A. C. (1984).Theory of Correspondences. Wiley & Sons, New York.
Klement, E. P., Puri, M. L., andRalescu, D. A. (1986). Limit theorems for fuzzy random variables.Proceedings of the Royal Society of London. Series A, 407:171–182.
Körner, R. (1997). On the variance of fuzzy random variables.Fuzzy Sets and Systems,92:83–93.
Krätschmer, V. (2001a).Induktive Statistik auf Basis unscharfer Meßkonzepte am Beispiel linearer Regressionsmodelle Postdoctoral Thesis, Faculty of Law and Economics, University of Saarland, Saarbrücken.
Krätschmer, V. (2001b). A unified approach to fuzzy random variables.Fuzzy Sets and Systems, 123:1–9.
Krätschmer, V. (2002a). Limit theorems for fuzzy-random variables.Fuzzy Sets and Systems, 126:253–263.
Krätschmer, V. (2002b). Some complete metrics on spaces of fuzzy subsets.Fuzzy Sets and Systems, 130:357–365.
Krätschmer, V. (2004). Probability theory in fuzzy sample spaces.Metrika, 60:167–189.
Ledoux, M. andTalagrand, M. (1991).Probability in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York.
López-Díaz, M. andGil, M. A. (1997). Constructive definitions of fuzzy random variables.Statistics & Probability Letters, 36:135–143.
López-Díaz, M. andGil, M. A. (1998). Approximating integrably bounded fuzzy random variables in terms of the generalized Hausdorff metric.Information Sciences, 104:279–291.
Musiał, K. (1991). Topics in the theory of Pettis integration.Rendiconti dell'Istituti di Matematica dell'Universitá di Trieste, 23:177–262.
Musiał, K. (1995). A few remarks concerning the strong law of large numbers for non-separable Banach space valued functions.Rendiconti dell'Istituti di Matematica dell'Universitá di Trieste, 26, suppl.:221–242.
Näther, W. (2000). On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data.Metrika, 51:201–221.
Puri, M. L. andRalescu, D. A. (1986). Fuzzy random variables.Journal of Mathematical Analysis and Applications, 114:409–422.
Stojakovic, M. (1994). Fuzzy random variables, expectation, and martingales.Journal of Mathematical Analysis and Applications, 184:594–606.
Talagrand, M. (1984). Pettis integral and measure theory.Memoirs of the American Mathematical Society, 51(307):ix +224pp.
Terán, P. (2003). A strong law of large numbers for random upper semicontinuous functions under exchangeability conditions.,Statistics & Probability Letters, 65:251–258.
Vakhania, N. N., Tarieladze, V. I., andChobanyan, S. A. (1987).Probability Distributions on Banach Spaces, Reidel, Dordrecht.
Vitale, R. A. (1985) L p —metrics for compact, convex sets.Journal of Approximation Theory, 45:280–287.
Zadeh, L. A. (1975). The concept of a linguistic variable and its applications to approximate reasoning I.Information Sciences, 8:199–249.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krätschmer, V. Integrals of random fuzzy sets. Test 15, 433–469 (2006). https://doi.org/10.1007/BF02607061
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02607061
Key Words
- Random fuzzy sets
- integrably bounded random fuzzy sets
- Aumann-integral
- Pettis-integral
- Bochner-integral
- dominated convergence theorems
- strong law of large numbers
- central limit theorems