Skip to main content
SpringerLink
Log in
Book cover

Advances in Mathematical Economics Volume 17 pp 1–37Cite as

Law of large numbers and Ergodic Theorem for convex weak star compact valued Gelfand-integrable mappings

  • Chapter
  • First Online:

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 17))

Abstract

We prove several results in the integration of convex weak star (resp. norm compact) valued random sets with application to weak star Kuratowski convergence in the law of large numbers for convex norm compact valued Gelfand-integrable mappings in the dual of a separable Banach space. We also establish several weak star Kuratowski convergence in the law of large numbers and ergodic theorem involving the subdifferential operators of Lipschitzean functions defined on a separable Banach space, and also provide an application to a closure type result arisen in evolution inclusions.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Notes

  1. 1.

    Namely \(\sum _{k=1}^{n}1_{A_{k}}g_{k}\) is a convex combination with positive rational coefficients of functions in S X 1(ℱ X ).

  2. 2.

    For more details, one may consult the proof of Theorem 5.5 below.

References

  1. Akhiat, F., Castaing, C., Ezzaki, F.: Some various convergence results for multivalued martingales. Adv. Math. Econ. 13, 1–33 (2010)

    Article  Google Scholar 

  2. Balder, E.J., Sambuccini, A.R.: Fatou’s lemma for multifunctions with unbounded values in a dual space. J. Convex. Anal. 12, 383–395 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Benabdellah, H., Castaing, C.: Weak compactness and convergences in L E′ 1[E]. Adv. Math. Econ. 3, 1–44 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bounkhel, M., Castaing, C.: State dependent sweeping process in Banach spaces. Set Valued Var. Anal. 20(2), 187–201 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourras, A., Castaing, C., Guessous, M.: Olech types lemma and Visentin-types theorem in Pettis integration and L E′ 1. In: Nishizawa, K. (ed.) NLA98: Convex Analysis and Chaos. The Third Symposium on Nonlinear Analysis, Josai University, 23–25, 1–26 July 1998

    Google Scholar 

  6. Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6(2), 359–374 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Castaing, C., Ezzaki, F., Hess, Ch.: Convergence of conditional expectation for unbounded closed convex random sets. Studia Math. 124(2), 133–148 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Castaing, C., Ezzaki, F., Lavie, M., Saadoune, M.: Weak star convergence of martingales in a dual space. In: Proceedings of the 9-th Edition of the International Conference on Function Spaces, Krakow, Poland. Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warsawa (2011)

    Google Scholar 

  9. Castaing, C., Hess, Ch., Saadoune, M.: Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions. Adv. Math. Econ. 11, 11–44 (2008)

    Article  Google Scholar 

  10. Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10, 1–20 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Castaing, C., Lavie, M.: Some applications of the Birkhoff-Kingman ergodic theorem. Adv. Math. Econ. 16, 1–38 (2012)

    Article  MathSciNet  Google Scholar 

  12. Castaing, C., Raynaud de Fitte, P.: \(\mathfrak{S}\)-uniform scalar integrability and strong law of large numbers for Pettis-integrable functions with values in a separable locally convex space. J. Theor. Probab. 13(1), 93–134 (2000)

    Google Scholar 

  13. Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer, Dordrecht (2004)

    Google Scholar 

  14. Castaing, C., Quang, N.V., Giap, D.X.: Various convergence results in strong law of large numbers for double array of random sets in Banach spaces. J. Nonlinear Convex Anal. 13(1), 615–636 (2011)

    Google Scholar 

  15. Castaing, C., Quang, N.V., Thuan, N.T.: A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers. Stat. Probab. Lett. 82, 83–95 (2012)

    Article  MathSciNet  Google Scholar 

  16. Castaing, C., Saadoune, M.: Convergences in a dual space with applications to Fatou Lemma. Adv. Math. Econ. 12, 23–69 (2009)

    Article  MathSciNet  Google Scholar 

  17. Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. In: Lecture Notes, vol. 580. Springer, Berlin (1977)

    Google Scholar 

  18. Christensen, J.P.R. (ed.): Topology and Borel Structure, Math. Studies 10, Notas de Mathematica, North-Holland, Amsterdam/American Elsevier, NY (1974)

    Google Scholar 

  19. Choirat, C., Hess, C., Seri, R.: A functional version of the Birkhoff ergodic theorem for a normal integrand: a variational approach. Ann. Probab. 1, 63–92 (2003)

    MathSciNet  Google Scholar 

  20. Clarke, F.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 274–262 (1975)

    Article  Google Scholar 

  21. Clarke, F.H., Stern, R.J., Wolenski, R.J.: Proximal smoothness and the lower C2 property. J. Convex Anal. 2, 117–144 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Csorgo, S., Tandori, K., Totik, V.: On the strong of large numbers for pairwise independent random variables. Acta Math. Hung. 42, 319–330 (1983)

    Article  MathSciNet  Google Scholar 

  23. Cuesta, J.A., Matrán, C.: Strong laws of large numbers in abstract spaces via Skorohod’s representation theorem. Sankhyā Ser. A 48(1), 98–103 (1986)

    Google Scholar 

  24. Cuesta, J.A., Matrán, C.: Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions. J. Multivar. Anal. 25(2), 311–322 (1988)

    Article  MATH  Google Scholar 

  25. Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel. Chapitres I à IV. Hermann, Paris (1975)

    MATH  Google Scholar 

  26. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

    MATH  Google Scholar 

  27. Edmond, J.F.: Problèmes d’évolution associés à des ensembles prox-reguliers. Inclusions et intégration de sous-différentiels, Thèse, Université des Sciences et Techniques du Languedoc, UER de Mathématiques, Juin 2004

    Google Scholar 

  28. Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 55, 119–122 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fitzpatrick, S., Lewis, A.S.: Weak-star convergence of convex sets. J. Convex Anal. 13(3+4), 711–719 (2006)

    Google Scholar 

  30. Grothendiek, A.: Espaces Vectoriels Topologiques, 3rd edn. Publi. Soc. Mat. Sao Paulo (1964)

    Google Scholar 

  31. Hess, C.: Loi de probabilité et indépendence des ensembles aléatoires à valeurs fermées dans un espace de Banach, Séminaire Analyse Convexe, Montpellier, Exposé no 7 (1983)

    Google Scholar 

  32. Hess, C.: Loi forte des grands nombres pour des ensembles alétoires non bornées à valeurs dans un espace de Banach séparable. C. R. Acad. Sci. Paris Ser. I 300, 177–180 (1985)

    MathSciNet  MATH  Google Scholar 

  33. Hess, C.: Multivalued strong laws of large numbers in the Slice topology. Application to integrands. Set Valued Anal. 2, 183–205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hess, C.: The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces. J. Convex Anal. 6(1), 163–182 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Hiai, F.: Convergence of conditional expectation and strong law of large numbers for multivalued random variables. Trans. Am. Math. Soc. 291(2), 613–627 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Hoffmann-Jørgensen, Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab. 4, 587–599 (1976)

    Google Scholar 

  38. Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen 42(1), 209–216 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Monteiro Marques, M.D.P.: Differential inclusions in nonsmooth mechanical problems, shocks and dry friction, progress in nonlinear differential equations and their applications. Birkhauser, Basel-Boston-Berlin 9 (1993)

    Google Scholar 

  40. Moreau, J.J.: Evolution problem asssociated with a moving convex set in a Hilbert space. J. Differ. Equat. 26, 347–374 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  41. Raynaud de Fitte, P.: Deux lois des grands nombres pour les ensembles aléatoires, Séminaire d’Analyse Convexe, Montpellier, exposé, No 1 (1991)

    Google Scholar 

  42. Shapiro, A., Xu, H.: Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions. J. Math. Anal. Appl. 325, 1390–1399 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Terán, P.: On consistency of stationary points of stochastic optimization problems in a Banach space. J. Math. Anal. Appl. 363, 569–578 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Thibault, L.: Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier (1976)

    Google Scholar 

  45. Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equat. 193, 1–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Valadier, M.: Contribution ł’Analyse Convexe, Thesis Paris (1970)

    Google Scholar 

  47. Valadier, M.: On conditional expectation of random sets. Annali di Matematica 126(1):81–91 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  48. Valadier, M.: Conditional expectation and ergodic theorem for a positive integrand. J. Nonlinear Convex Anal. 1, 233–244 (2002)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Montpellier II, Case courrier 051, 34095, Montpellier Cedex 5, France

    C. Castaing

  2. Laboratoire Raphaël Salem, UMR CNRS 6085, UFR Sciences, Université de Rouen, Avenue de l’Université, 12, 76801, Saint Etienne du Rouvray, France

    P. Raynaud de Fitte

Authors
  1. C. Castaing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Raynaud de Fitte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Castaing .

Editor information

Editors and Affiliations

  1. Graduate School of, Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo, 153-8914, Japan

    Shigeo Kusuoka

  2. Dept. Economics, Keio University, Mita 2-15-45, Tokyo, 108-8345, Japan

    Toru Maruyama

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Japan

About this chapter

Cite this chapter

Castaing, C., de Fitte, P.R. (2013). Law of large numbers and Ergodic Theorem for convex weak star compact valued Gelfand-integrable mappings. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 17. Advances in Mathematical Economics, vol 17. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54324-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-54324-4_1

  • Received:

  • Revised:

  • Published:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-54323-7

  • Online ISBN: 978-4-431-54324-4

  • eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)

Publish with us

Policies and ethics

Search

Navigation