Download PDF - Convergence of conditional expectations for unbounded closed convex random setsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Convergence of conditional expectations for unbounded closed convex random sets

Authors 1, 2, 3

Affiliations

  1. Département de Mathématiques, Case 051, Université Montpellier II, F-34095 Montpellier Cedex 5, France
  2. Département de Mathématiques Mohammed V, Faculté des Sciences Agdal, Rabat, Maroc F-34095 Montpellier Cedex 5, France
  3. CEREMADE, URA CNRS No 749, Université Paris-Dauphine, 75775 Paris Cedex 16, France

Abstract

We discuss here several types of convergence of conditional expectations for unbounded closed convex random sets of the form EℬnXn where (ℬn) is a decreasing sequence of sub-σ-algebras and (Xn) is a sequence of closed convex random sets in a separable Banach space.

Bibliography

  1. [A] H. Attouch, Variational Convergence for Functions and Operators, Appl. Math. Ser., Pitman, Boston, 1984.
  2. [CE1] C. Castaing and F. Ezzaki, Variational inequalities for integrand martingales and additive random sequences, Acta Math. Vietnam. 18 (1993), 137-171.
  3. [CE2] C. Castaing and F. Ezzaki, SLLN for convex random sets and random lower semicontinuous integrands, preprint, Département de Mathématiques, Université Montpellier II, 1995.
  4. [CV] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, 1977.
  5. [Ch] A. Choukairi-Dini, Contribution à l'étude des martingales et au lemme de Fatou multivoque, Thèse, Université Montpellier II, 1988.
  6. [Co] J. Couvreux, Etude de problèmes de convergence et d'approximation de fonctionnelles intégrales et d'espérances conditionnelles multivoques, Thèse, Université Paris-Dauphine, 1995.
  7. [Ez1] F. Ezzaki, Contributions aux problèmes de convergence des suites adaptées et des ensembles aléatoires, Thèse, Université Montpellier II, 1993.
  8. [Ez2] F. Ezzaki, A general dominated convergence theorem for unbounded random sets, Bull. Polish Acad. Sci. Math. 44 (1996), 353-361.
  9. [He1] C. Hess, Lemme de Fatou et convergence dominée pour les ensembles aléatoires non bornés et des intégrandes, Sém. Analyse Convexe Montpellier, exp. no. 8, 1986.
  10. [He2] C. Hess, Convergence of conditional expectations for unbounded random sets, integrands, and integral functionals, Math. Oper. Res. 16 (1991), 627-649.
  11. [He3] C. Hess, Mesurabilité, convergence et approximation des multifonctions à valeurs dans un e.l.c.s., Sém. Analyse Convexe Montpellier, exp. no. 9, 1985.
  12. [He4] C. Hess, Measurability and integrability of the weak upper limit of a sequence of multifunctions, J. Math. Anal. Appl. 153 (1990), 226-229.
  13. [He5] C. Hess, Multivalued strong law of large numbers in the slice topology. Application to integrands, Set-Valued Anal. 2 (1994), 183-205.
  14. [Hi1] F. Hiai, Multivalued conditional expectations, multivalued Radon-Nikodym theorems, integral representation of additive operators and multivalued strong laws of large numbers, preprint, Conference of Catania, 1983.
  15. [Hi2] F. Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985), 613-627.
  16. [HU] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182.
  17. [Mo] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 151-155.
  18. [Ne] J. Neveu, Martingales à temps discret, Masson, 1972.
  19. [SZ] D. Szynal and W. Zięba, On some characterization of almost sure convergence, Bull. Polish Acad. Sci. Math. 34 (1986), 635-642.
  20. [We] R. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, in: Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions (eds.), Wiley, Chichester, 1980, 375-403.
  21. [Zi] W. Zięba, On the L1_ℱ convergence of conditional amarts, J. Multivariate Anal. 26 (1988), 104-110.
Studia Mathematica
1997/124/2
Export to PBN, Opens in new tab
Pages:
133-148
Main language of publication
English
Received
1996-04-16
Published
1997
Exact and natural sciences