Skip to main content
SpringerLink
Log in

Capacities, Set-Valued Random Variables and Laws of Large Numbers for Capacities

  • Chapter
Book cover Integrated Uncertainty Management and Applications

Part of the book series: Advances in Intelligent and Soft Computing ((AINSC,volume 68))

  • 970 Accesses

Abstract

In this paper, we shall survey some connections between the theory of set-valued random variables and Choquet theory. We shall focus on investigating some results of the relationships between the distributions of set-valued random variables and capacities, and also some connections between the Aumann integral and the Choquet integral. Then we shall review some results on laws of large numbers (LLN’s) for set-valued random variables and for capacities, and point out some relations between these two kinds of LLN’s. Finally we shall give a new strong LLN of exchangeable random variables for capacities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allais, M.: Le comportement de l’homme rationnel devant le risque: Critique des postulates et axiomes de l’ecole americaine. Econometrica 21, 503–546 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  2. Artstein, Z., Hansen, J.C.: Convexification in limit laws of random sets in Banach spaces. Ann. Probab. 13, 307–309 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Ann. Probab. 3, 879–882 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aumann, R.: Integrals of set valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aumann, R.J.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  6. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lect. Notes in Math., vol. 580. Springer, Berlin (1977)

    MATH  Google Scholar 

  7. Castaldo, A., Marinacci, M.: Random correspondences as bundles of random variables. In: The 2nd International Symposium on Imprecise Probability and Their Applications, Ithaca, New York (2001)

    Google Scholar 

  8. Castaldo, A., Maccheroni, F., Marinacci, M.: Random correspondences as bundles of random variables. Sankhya 66, 409–427 (2004)

    MathSciNet  Google Scholar 

  9. Chen, Z., Kulperger, R.: Minmax pricing and Choquet pricing. Insurance: Mathematics and Economics 38, 518–528 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Choquet, G.: Theory of capacities. Ann Inst. Fourier 5, 131–295 (1953)

    MathSciNet  Google Scholar 

  11. Colubi, A., López-Díaz, M., Dominguez-Menchero, J.S., Gil, M.A.: A generalized strong law of large numbers. Probab. Theory and Rel. Fields 114, 401–417 (1999)

    Article  MATH  Google Scholar 

  12. Couse, I., Sanchez, L., Gil, P.: Imprecise distribution function associated to a random set. Information Sci. 159, 109–123 (2004)

    Article  Google Scholar 

  13. Dempster, A.: Upper and lower probabilities induced by multivalued mapping. Ann. Math. Statist. 38, 325–339 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  14. Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publishers, Boston (1994)

    MATH  Google Scholar 

  15. Ellsberg, D.: Risk, ambiguity, and the Savage axioms. Quart. J. Econom. 75, 643–669 (1961)

    Article  Google Scholar 

  16. Epstein, L., Schneider, M.: IID: independently and indistinguishably distributed. J. Econom. Theory 113, 32–50 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fu, K., Zhang, L.: Strong limit theorems for random sets and fuzzy random sets with slowly varying weights (2007) (to appear)

    Google Scholar 

  18. Graf, S.: A Radon-Nikodym theorem for capacites. J. Math. 320, 192–214 (1980)

    MATH  MathSciNet  Google Scholar 

  19. Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals: Theory and Applications. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  20. Guan, L., Li, S.: Laws of large numbers for weighted sums of fuzzy set-valued random variables. International J. of Uncertainty Fuzziness and Knowledge-Based Systems 12, 811–825 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Guan, L., Li, S., Ogura, Y.: A strong law of large numbers of fuzzy set-valued random variables with slowly varying weights. International J. Automation and Control 2, 365–375 (2008)

    Article  Google Scholar 

  22. Hiai, F.: Strong laws of large numbers for multivalued random variables, Multifunctions and Integrands. In: Salinetti, G. (ed.). Lecture Notes in Math., vol. 1091, pp. 160–172. Springer, Berlin (1984)

    Google Scholar 

  23. Hiai, F., Umegaki, H.: Integrals, conditional expectations and martingales of multivalued functions. J. Multiva. Anal. 7, 149–182 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton (1974)

    MATH  Google Scholar 

  25. Inoue, H.: A strong law of large numbers for fuzzy random sets. Fuzzy Sets and Syst. 41, 285–291 (1991)

    Article  MATH  Google Scholar 

  26. Giné, E., Hahn, G., Zinn, J.: Limit theorems for random sets: an application of probability in Banach space results. Lect. Notes in Math., vol. 990, pp. 112–135 (1983)

    Google Scholar 

  27. Klement, E.P., Puri, L.M., Ralescu, D.A.: Limit theorems for fuzzy random variables. Proc. Roy. Soc. Lond. 407, 171–182 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  28. Li, S., Ogura, Y.: Convergence of set valued sub- and super-martingales in the Kuratowski–Mosco sense. Ann. Probab. 26, 1384–1402 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. Li, S., Ogura, Y.: Strong laws of numbers for independent fuzzy set-valued random variables. Fuzzy Sets Syst. 157, 2569–2578 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Li, S., Ogura, Y., Kreinovich, V.: Limit Theorems and Applications of Set-Valued and Fuzzy Sets-Valued Random Variables. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  31. Maccheroni, F., Marinacci, M.: A strong law of large numbers for capacites. Ann. Probab. 33, 1171–1178 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Marinacci, M.: Limit laws for non-additive probabilities and their frequentist interpretation. J. Rconom. Theory 84, 145–195 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nguyen, H.T.: On random sets and belief functions. J. Math. Anal. Appl. 65, 531–542 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nguyen, H.T.: An Introduction to Radom Sets. CRC Press, Boca Raton (2007)

    Google Scholar 

  35. Puri, M.L., Ralescu, D.A.: Strong law of large numbers for Banach space valued random sets. Ann. Probab. 11, 222–224 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rebille, Y.: Law of large numbers for non-additive measures. arXiv:0801.0984v1 [math. PR] (January 7, 2008)

    Google Scholar 

  37. Sarin, R., Wakker, P.: A simple axiomatization of nonadditive expected utility. Econometrica 60, 1255–1272 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  38. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  39. Shafer, G.: A methematical Theory of Evidence. Princeton University Press, Princeton (1976)

    Google Scholar 

  40. Song, Y., Yan, J.-A.: Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders (2007) (Preprint)

    Google Scholar 

  41. Sugeno, M.: Theory of fuzzy integrals and its applications, Ph. D. dissertation, Tokyo Institute of Technology (1974)

    Google Scholar 

  42. Taylor, R.L., Inoue, H.: A strong law of large numbers for random sets in Banach spaces. Bull. Instit. Math. Academia Sinica 13, 403–409 (1985)

    MATH  MathSciNet  Google Scholar 

  43. Taylor, R.L., Inoue, H.: Convergence of weighted sums of random sets. Stoch. Anal. Appl. 3, 379–396 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  44. Taylor, R.L., Inoue, H.: Laws of large numbers for random sets. In: Random Sets, Minneapolis, MN, pp. 347–360 (1996); IMA Vol. Math. Appl. vol. 97. Springer, New York (1997)

    Google Scholar 

  45. Wakker, P.: Testing and Characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica 57, 571–587 (2001)

    MathSciNet  Google Scholar 

  46. Wang, S.: A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance 67, 15–36 (2000)

    Article  Google Scholar 

  47. Zhang, J.: Subjective ambiguity, expected utility and Choquet expected utility. Econom. Theory 20, 159–181 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  48. Zhang, W., Li, S., Wang, Z., Gao, Y.: An Introduction of Set-Valued Stochastic Processes. Sinence Press, Beijing (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, P.R. China

    Shoumei Li & Wenjuan Yang

Authors
  1. Shoumei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenjuan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Knowledge Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Ishikawa, Japan

    Van-Nam Huynh  & Yoshiteru Nakamori  & 

  2. Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, BS8 1TR, Bristol, United Kingdom

    Jonathan Lawry

  3. Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

    Masahiro Inuiguchi

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Li, S., Yang, W. (2010). Capacities, Set-Valued Random Variables and Laws of Large Numbers for Capacities. In: Huynh, VN., Nakamori, Y., Lawry, J., Inuiguchi, M. (eds) Integrated Uncertainty Management and Applications. Advances in Intelligent and Soft Computing, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11960-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11960-6_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11959-0

  • Online ISBN: 978-3-642-11960-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation