Limit theorems for random sets: An application of probability in banach space results

  • Conference paper
  • First Online:
Probability in Banach Spaces IV

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 990))

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
EUR 26.99
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 37.40
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** 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

  • Araujo, A. and Giné, E. (1980). The central limit theorem for real and Banach valued random variables. Wiley, New York.

    MATH  Google Scholar 

  • Arrow, K.J. and Hahn, F.H. (1971). General Competitive Analysis. Holden-Day, San Francisco.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Aumann, R.J. (1965). Integrals of set-valued functions. Indiana Math. J. 24, 433–441.

    MathSciNet  MATH  Google Scholar 

  • Billingsley, P. (1968). Convergence of probability measures. Wiley, New York.

    MATH  Google Scholar 

  • Byrne, C.L. (1978). Remarks on the set-valued integrals of Debreu and Aumann. J. Math. Analysis and Appl. 62, 243–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Cressie, N. (1978). A strong limit theorem for random sets. Adv. Appl. Prob. Suppl. 10, 36–46.

    Article  MathSciNet  MATH  Google Scholar 

  • Cressie, N. (1979). A central limit theorem for random sets. Z. Wahrscheinlichkeitstheorie 49, 37–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Debreu, G. (1966). Integration of correspondences. Proc. Fifth Berkeley Symp. Math. Statist. and Probability 2, 351–372. Univ. of California Press.

    MathSciNet  Google Scholar 

  • Garling, D.J.H. and Gordon, Y. (1971). Relations between some constants associated with finite dimensional Banach spaces. Israel J. Math. 9, 346–361.

    Article  MathSciNet  MATH  Google Scholar 

  • Giné, E. (1976). Bounds for the speed of convergence in the central limit theorem in C(S). Z. Wahrscheinlichkeitstheorie 36, 317–331.

    Article  MathSciNet  MATH  Google Scholar 

  • Giné, E. (1980). Sums of independent random variable and sums of their squares. Pub. Mat. UAB No 22, Actes VII JMHL.

    Google Scholar 

  • Giné, E. and Marcus, M.B. (1981). On the central limit theorem in C(K). In "Aspects statistiques et aspects physiques des processus Gaussiens", 361–383. Colloques Intern of the CNRS no 307, CNRS, Paris.

    Google Scholar 

  • Hormander, L. (1954). Sur la fonction d'appui des ensembles convexes dans un espace localement convexe. Arkiv för Matematik 3, 181–186.

    Article  MathSciNet  MATH  Google Scholar 

  • Jain, N.C. and Marcus, M.B. (1975). Central limit theorem for C(S)-valued random variables. J. Funct. Anal. 19, 216–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Kendall, D.G. (1974). Foundations of a theory of random sets. In Stochastic Geometry, ed. E.F. Harding and D.G. Kendall, Wiley, New York.

    Google Scholar 

  • Kuratowski, K. and Ryll-Nardzewski, C. (1965). A general theory of selectors. Bull. Pol. Acad. Sci. 13, 397–403.

    MathSciNet  MATH  Google Scholar 

  • Lindenstrauss, J. and Tzafriri, L. (1979). Classical Banach Spaces, Vol. II: Function Spaces. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.

    MATH  Google Scholar 

  • Mityagin, B.S. (1961). Approximate dimension and bases in nuclear spaces. Russian Math. Surveys 16, 59–127.

    Article  MathSciNet  MATH  Google Scholar 

  • Mourier, E. (1955). L-random elements and L*-random elements in Banach spaces. Proc. Third Berkeley Symp. Math. Statist. and Probability 2, 231–242. Univ. of California Press.

    MathSciNet  Google Scholar 

  • Pisier, G. (1975). Le Théorème de la limite centrale et la loi due logarithme itéré dans les espace de Banach. Séminaire Maurey-Schwartz 1975–76, Exposés 3 et Ecole Polytechnique, Paris.

    Google Scholar 

  • Pisier, G. (1979). Some applications of the complex interpolation method to Banach lattices. Journal d'Analyse Math. 35, 264–281.

    Article  MathSciNet  MATH  Google Scholar 

  • Pisier, G. and Zinn, J. (1978). On the limit theorems for random variables with values in the spaces Lp, p ≥ 2. Z. Wahrscheinlichkeitstheorie 41, 289–304.

    Article  MathSciNet  MATH  Google Scholar 

  • Puri, M.L. and Ralescu, D.A. (1982). Strong law of large numbers for Banach space valued random sets. To appear in Ann. Probability, February, 1983.

    Google Scholar 

  • Robbins, H.E. (1944). On the measure of a random set. Ann. Math. Statist. 14, 70–74.

    Article  MathSciNet  MATH  Google Scholar 

  • Robbins, H.E. (1945). On the measure of a random set, II. Ann. Math. Statist. 15, 342–347.

    Article  MathSciNet  MATH  Google Scholar 

  • Rvačeva, E.L. (1962). On domains of attraction of multidimensional distributions. Selected Translations in Math. Stat. and Probability v. 1, 183–205.

    Google Scholar 

  • Trader, D.A. and Eddy, W.F. (1981). A central limit theorem for Minkowski sums of random sets. Carnegie-Mellon University Technical Report No. 228.

    Google Scholar 

  • Vitale, R.A. (1981). A central limit theorem for random convex sets. Technical Report Claremont Graduate School.

    Google Scholar 

  • Weil, W. (1982). An application of the central limit theorem for Banach space-valued random variables to the theory of random sets. Z. Wahrscheinlichkeitstheorie 60, 203–208.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Anatole Beck Konrad Jacobs

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Giné, E., Hahn, M.G., Zinn, J. (1983). Limit theorems for random sets: An application of probability in banach space results. In: Beck, A., Jacobs, K. (eds) Probability in Banach Spaces IV. Lecture Notes in Mathematics, vol 990. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064267

Download citation

  • DOI: https://doi.org/10.1007/BFb0064267

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12295-1

  • Online ISBN: 978-3-540-39870-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Navigation