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A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values

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Abstract

Given a nondecreasing sequence (ℬ n ) of sub-σ-fields and a real or vector valued random variable f, the Lévy Martingale convergence Theorem (LMCT) asserts that E(f/ℬ n ) converges to E(f/ℬ∞) almost surely and in L 1, where ℬ∞ stands for the σ-field generated by the ℬ n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of ℱ(X), the space of nonempty closed sets of a Banach space X, when ℱ(X) is endowed either with the Painlevé–Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/ℬ n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.

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REFERENCES

  1. Aubin, J. P. (1984). L'analyse Nonlinéaire et ses Motivations Économiques, Masson, Paris.

    Google Scholar 

  2. Attouch, H. (1984). Variational convergence for functions and operators. Appl. Math. Series, Pittman advanced publishing program.

  3. Balder, E. J., and Hess, C. (1995). On the unbounded multivalued version of Fatou's lemma. Math. Oper. Res. 20(1).

  4. Balder, E. J., and Hess, C. (1996). Two generalizations of Komlos' theorem with lower closure-type application. J. Convex Anal. 3, 1–20.

    Google Scholar 

  5. Barbati, A., and Hess, C. (1998). On the largest class of closed convex valued multifunctions for which Effros measurability and scalar measurability coincide. Set-Valued Anal. 6, 209–236.

    Google Scholar 

  6. Beer, G. (1990). On Mosco convergence of convex sets. Bull. Austral. Math. Soc. 38, 239–253.

    Google Scholar 

  7. Beer, G. (1991). Topologies on closed and closed convex sets and Effros-measurability of set valued functions, Séminaire d'Analyse Convexe de l'Univesité de Montpellier, Exposé No. 2.

  8. Beer, G. (1993). Topologies on closed and closed convex sets, Mathematics and Its Applications, Vol. 268, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  9. Castaing, C., Ezzaki, F., Hess, C. (1996). Convergence of conditional expectations for unbounded closed convex random sets, Preprint. To appear in Studia Mathematica.

  10. Castaing, C., and Valadier, M. (1977). Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, Vol. 580, Springer.

  11. Choquet, G. (1947). Convergences. Annales de l'Université de Grenoble 23, 55–112.

    Google Scholar 

  12. Choukairi-Dini, A. (1987). Sur la convergence d'espérances conditionnelles de multiapplications, Séminaire d'Analyse Convexe de l'Université de Montpellier, Exposé No. 10.

  13. Costé, A. (1980). Sur les martingales multivoques. C. R. Acad. Sci. Paris, Série A 290, 953–956.

    Google Scholar 

  14. Couvreux, J. (1995). Etude de problèmes de convergence et d'approximation de fonctionelles intégrales, et d'espérances conditionnelles multivoques, Thèse, Ceremade, Université Paris Dauphine.

  15. Dal Maso, G. (1993). An Introduction to γ-Convergence, Birkhäuser.

  16. Dal Maso, G., and Mokdica, L. (1986). Nonlinear stochastic homogenization. Ann. Mat. Pura ed Appl. 144, 347–389.

    Google Scholar 

  17. De Giorgi, E. (1979). Convergence problems for functions and operators. Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, pp. 131–188.

  18. Dong, W. L., and Wang, Z. P. (Undated). On the representation and regularity of continuous parameter multivalued martingales, to appear in Trans. Amer. Math. Soc.

  19. Egghe, L. (1985). Stopping Times Techniques in Analysis, Universiteit Antwerpen, Universitaire Instelling Antwerpen, Department wiskunde informatica, and Cambridge University Press, Cambridge.

    Google Scholar 

  20. Ezzaki, F. (1993). Thèse, Université de Montpellier.

  21. Ezzaki, F. (1996). Thèse de Doctorat d'Etat, Université Mohammed V, Faculté des Sciences de Rabat, Maroc.

  22. Hess, C. (1985). Mesurabilité, convergence and approximation des multifonctionsà valeurs dans un e.l.c.s., Séminaire d'Analyse Convexe de l'Université de Montpellier, Exposé No. 9.

  23. Hess, C. (1986a). Quelques résultats sur la mesurabilité des multifonctions à valeurs dans un espace métrique séparable, Séminaire d'analyse Convexe de l'Université de Montpellier, Exposé No. 1.

  24. Hess, C. (1986b). Lemme de Fatou et théoreme de la convergence dominée pour les ensembles aleatoires non bornés, Séminaire d'Analyse Convexe de l'Université de Montpellier, Exposé No. 8.

  25. Hess, C. (1990). Measurability and integrability of the weak upper limit of a sequence of multifunctions. J. Math. Anal. Appl. 153(1), 226–249.

    Google Scholar 

  26. Hess, C. (1991a). Sur l'existence de selections martingales et la convergence des surmartingales multivoques. C. R. Acad. des Sci. Paris, Série I 312, 149–154.

    Google Scholar 

  27. Hess, C. (1991b). On multivalued martingales whose values may be unbounded: Martingale selectors and Mosco convergence. J. of Multivariate Anal. 39(1), 175–201.

    Google Scholar 

  28. Hess, C. (1991c). Convergence of conditional expectations for unbounded random sets, integrands and integral function. Math. Oper. Res. 16(3), 627–649.

    Google Scholar 

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

    Google Scholar 

  30. Hess, C., (1996). Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator. Ann. Stat. 24(3), 1298–1315.

    Google Scholar 

  31. Hiai, F. (1983). Multivalued conditional expectations, multivalued Radon-Nykodym theorem, integral representation of additive operators, and multivalued strong, laws of large numbers, Conference of Catania.

  32. Hiai, F. (1984). Strong laws of large numbers for multivalued random variables, in Lecture Notes in Math., Vol. 1091, Springer.

  33. Hiai, F. (1983). Convergence of conditional expectations and strong laws of large numbers for multivalued random variables. Trans. Amer. Math. Soc. 291(2), 613–627.

    Google Scholar 

  34. Hiai, F. and Umegaki, H. (1977). Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7(1), 149–168.

    Google Scholar 

  35. Kleï, H. A. (1988). A compactness criterion in L 1(E) and Radon-Nikodym Theorems for multimeasure. Bull. Sc. Math., 2e Série 112, 305–324.

    Google Scholar 

  36. Levi, S., Lucchetti, R., and Pelant, J. (1993). On the infimum of the Hausdorff and Vietoris topologies. Proc. AMS 118(3), 971–978.

    Google Scholar 

  37. Lévy, P. (1935). Propriétés asymptotiques des sommes de variables aléatoires enchaînées. Bull. Sci. Math. 59(2), 84–96; 109–128.

    Google Scholar 

  38. Lévy, P. (1937). Théorie de l'Addition des Varaibles Aléatoires, Gauthier-Villars, Paris.

    Google Scholar 

  39. Meyer, P. A. (1966). Probabilités et Potentiel, Hermann, Paris.

    Google Scholar 

  40. Moreau, J. J. (1996). Fonctionnelles Convexes, Séminaire sur leséquations aux dérivées partielles, Collège de France.

  41. Mosco, U. (1969). Convergence of convex solutions of variational inequalities. Adv. Math. 3, 510–585.

    Google Scholar 

  42. Neveu, J. (1972a). Martingales à Temps Discret, Masson, Paris.

    Google Scholar 

  43. Neveu, J. (1972b). Convergence presque sûre des martingales multivoques. Ann. Inst. Henri Poincaré VIII(1).

  44. Papageorgiou, N. S. (1995) On the conditional expectation and convergence properties of random sets. Trans. Amer. Math. Soc. 347(7).

  45. Piccinini, L. (1996). Thèse, Université de Montpellier.

  46. Raynaud de Fitte, P. (1990). Thèse, Université de Montpellier.

  47. Rockafellar, R. T., and Wets, R. J. B. (1984). Variational Systems, an Introduction. In G. Salinetti (ed.), Multifunctions and integrands, Stochastic Analysis, Approximation and Optimization, Lecture Notes in Math., Springer-Verlag, Berlin, Vol. 1091, pp. 1–54.

    Google Scholar 

  48. Sonntag, Y. and Zalinescu, C. (1990). Set convergences: An attempt of classification. Proc. Intl. Conf. on Diff. Equations and Control Theory, Iasi, Romania.

  49. Valadier, M. On Conditional Expectations of Random Sets. Anns. Mat. Purae Appl. (IV) CXXVI, 81–91.

  50. Valadier, M. (1980). Sur l'espérance conditionnelle multivoque non convexe. Ann. Inst. Henri Poincaré 16, 109–116.

    Google Scholar 

  51. Wang, Z. P., and Xue, X. H. (1994). On convergence and closedness of multivalued martingales. Trans. Amer. Math. Soc. 341(2).

  52. Wijsman, R. A. (1964). Convergence of sequences of convex sets, cones and functions. Bull. AMS 70, 186–188.

    Google Scholar 

  53. Wijsman, R. A. (1966). Convergence of sequences of convex sets II. Trans. Amer. 123, 32–45.

    Google Scholar 

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  1. Centre de Recherche Viabilité, Jeux, Contrôle, ERS 644 CNRS, Université Paris Dauphine, 75775, Paris Cedex 16, France

    Jérôme Couvreux & Christian Hess

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  1. Jérôme Couvreux
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  2. Christian Hess
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Couvreux, J., Hess, C. A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values. Journal of Theoretical Probability 12, 933–969 (1999). https://doi.org/10.1023/A:1021688919194

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