Traditional techniques to prove some limit theorems for fuzzy random variables

  • Ana Colubi
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 87)


In the last years, some limit theorems for fuzzy random variables have been proven by means of different techniques developed for this purpose. In this work we deal with the cadlag representation of a kind of fuzzy sets to show that these limit results can be also proved by applying well-known techniques in Probability Theory (specifically, the ones which make valid the analogous theorems for D[0, 1]-valued random elements). In this context, we will study a strong law of large numbers (whose proof will suggest a characterization of the uniform convergence) and a strong law of the iterated logarithm. Furthermore, we will check the relationships between these techniques and the ones used by Molchanov to prove a SLLN for the same random elements.


Limit Theorem Random Element Iterate Logarithm Fuzzy Random Variable Prove Limit Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artstein, Z. and Hansen, J.C. (1985). Convexification in limit laws of random sets in Banach spaces. Ann. Probab. 13, 307 - 309.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aubin, J.P. (1999). Mutational and Morphological Analysis. Birkhäuser, Boston.MATHCrossRefGoogle Scholar
  3. 3.
    Aumann, R.J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1 - 12.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Billingsley, P. (1968). Convergence of Probability Measures. John Wiley Sons, New York.MATHGoogle Scholar
  5. 5.
    Colubi, A., López-Díaz, M., Domínguez-Menchero, J.S. and Gil, M.A. (1997). A generalized strong law of large numbers. Technical Report. Universidad de Oviedo.Google Scholar
  6. 6.
    Colubi, A., López-Díaz, M., Domínguez-Menchero, J.S. and Gil, M.A. (1999). A generalized strong law of large numbers. Probab. Theory Relat. Fields 114, 401 - 417.MATHCrossRefGoogle Scholar
  7. 7.
    Colubi, A., Domínguez-Menchero, J.S., López-Díaz, M. and Ralescu D. A. (2001). A DE[0, 1] representation of random upper semicontinuous functions, (submitted for publication)Google Scholar
  8. 8.
    Colubi, A., Domínguez-Menchero, J.S., López-Díaz, M. and Körner R. (2001). A method to derive strong laws of large numbers for random upper semicontinuous functions, Statist. Probab. Let. (accepted for publication).Google Scholar
  9. 9.
    Daffer, P.Z. and Taylor, R.L. (1979). Laws of large numbers for D[0, 1]. Ann. Prob. 7, 85 - 95.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Daffer, P. and Schiopu-Kratina, I. (1988). L’tightness and the law of large numbers in D(IR). Can. J. Stat. 16, 393 - 397.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Debreu, G. (1967). Integration of correspondences. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 1965/66 2, Part 1. Univ. of California Press, Berkeley, 351 - 372.Google Scholar
  12. 12.
    Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes. Characterizations and Convergence.John Wiley Sons.Google Scholar
  13. 13.
    Giné, E., Hahn, M. and Zinn, J. (1983). Limit theorems for random sets: an application of probability in Banach space results. Probability in Banach spaces IV. Berlin, Springer-Verlag 990, 112 - 135.Google Scholar
  14. 14.
    Hiai, F. (1985). Convergence of conditional expectations and strong laws of large numbers for multivalued random variables. Trans. Amer. Math. Soc. 291, 613 - 627.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hoffman-Jorgensen, J. (1985a). The law of large numbers for non-measurable and non-separable random elements. Asterisque. 131 299 - 356.Google Scholar
  16. 16.
    Klement, E.P., Puri M.L. and Ralescu, D.A. (1986). Limit theorems for fuzzy random variables. Proc. R. Soc, Lond. A 407, 171 - 182.MathSciNetMATHGoogle Scholar
  17. 17.
    López-Díaz, M. and Gil, M.A. (1998a). Approximating integrably bounded fuzzy random variables in terms of the "generalized" Hausdorff metric. Inform. Sci. 104, 279 - 291.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lyashenko, N.N. (1982). Limit theorems for sums of independent compact random subsets of euclidean space. J. Soviet. Math. 20, 2187 - 2196.MATHCrossRefGoogle Scholar
  19. 19.
    Molchanov, I. (1999). On strong laws of large numbers for random upper semi-continuous functions. J. Math. Anal. Appl. 235, 349 - 355.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Proske, F. (1997). Grenzwertsätze far Fuzzy-Zufallsvariablen unter dem Gesichspunkt der Wahrscheinlichkeitstheorie ouf inseparablen semigruppen. PhD Thesis. Univ. Ulm.Google Scholar
  21. 21.
    Puri, M.L. and Ralescu, D.A. (1981). Différentielle d'une fonction floue. C.R. Acad. Sci. Paris Sér. A 293, 237 - 239.MathSciNetMATHGoogle Scholar
  22. 22.
    Puri, M.L. and Ralescu, D. (1986). Fuzzy random variables. J. Math. Anal. Appl. 114, 409 - 422.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ana Colubi
    • 1
  1. 1.Departamento de Estadística e I.O. y D.M.Universidad de OviedoOviedoSpain

Personalised recommendations