Abstract

Convergence of martingales, amarts, pramarts, mils, have been extentively studied in recent years by a lot of authors: Bellow [3], Blake [4], Chatterji [10], Castaing [5], Castaing and Ezzaki [8], Choukairi ([11], [12], [14]), Daures [15], Bagchi [1], Hiai and Umegaki [20], Hess [19], Egghe ([17], [18]), Millet and Sucheston [26], Lavie [22], Luu ([23], [24], [25]), Slaby [28], Derras [16], Neveu [27], Talagrand [29], and many others. New convergence results for bounded pramarts in LE1L^1_E and in the space Lwc(E)1L^1_{wc(E)} of integrably bounded multifunction with convex weakly compact values are presented. The main purpose of this paper is to obtain the Linear topology convergence (introduced by Beer [2]) and Mosco-convergence of multivalued pramarts.

Author information

Contact details are reproduced from the original publication and may be historical.

Ahmed Choukairi-Dini

Université Ibnou-Zohr, Faculté des Sciences, Departement de Maths, Agadir, Marocco

Keywords

  • multivalued conditional expectation
  • pramart
  • kadec-klee’s norm
  • Mosco topology
  • linear topology

References

  1. S. Bagchi: On a.s. convergence of class of multivalued asymptotic martingales. Ann. Inst. H. Poincaré, t. 21. No. 4 (1985).Google Scholar
  2. G. Beer: Support and distance functionals for convex sets. Numer. Funct. Analy. and Optimis. 10 (162) 15–30 (1989).https://doi.org/10.1080/01630568908816288Google Scholar
  3. A. Bellow: Uniform amarts: a class of asymptotic martingales for which strong almost sure convergence obtains. Z. Wahrsch. verw. Geb. 41, 177–191 (1978).https://doi.org/10.1007/BF00534238Google Scholar
  4. L.H. Blake: A generalisation of martingales and consequent convergence theorems. Pacifis J. Math. 35 (1970) 279–283.https://doi.org/10.2140/pjm.1970.35.279Google Scholar
  5. C. Castaing: Quelques résultats de convergence des suites adaptées. Séminaire d’Analyse Convexe. Montpellier, Expose no. 2 (1987).Google Scholar
  6. C. Castaing: Sur la décomposition de Slaby, applications aux problèmes de convergence en probabilites, èconmie mathématiques. Théorie du contrôle Minimisation. S.A.C. Montpellier. Expose no. 3 (1989).Google Scholar
  7. C. Castaing, P. Clauzure: Version vectorielle et multivoguè d’un résultat de Brooks-Chacon. S.A.C. Montpellier. Exposé no. 4 (1988).Google Scholar
  8. C. Castaing, F. Ezzaki: Some convergence results for multivalued martingales in the limit. S.A.C. Montpellier. Exposé no. 1 (1990).Google Scholar
  9. C. Castaing, M. Valadier: Convex Analysis and measurable multifunctions. Lecture notes in Maths 580. Springer, 1977.https://doi.org/10.1007/BFb0087685Google Scholar
  10. S.D. Chatterji: Martingale convergence and the Radon-Nikodym theorem in Banach space. Math. SCand. 22, 21–41.https://doi.org/10.7146/math.scand.a-10868Google Scholar
  11. A. Choukairi-Dini: M -Convergence et régularité des martingales multivoques. Epi-martingale. S.A.C. Montpellier (1987) et Journal of Multivariate Analysis. Vol. 33. No. 1 April (1990).https://doi.org/10.1016/0047-259X(90)90005-3Google Scholar
  12. A. Choukairi-Dini: M -Convergence des martingales. Application aux amarts uniformes et aux intégrandes (II). S.A.C. Montpellier (1989). Exposé no. 7. Et, C.R.A.S. Paris. t. 309, Série I (1989) 889–892.Google Scholar
  13. A. Choukairi-Dini: Sur la convergence d’esperance conditionnelle de multi applications. T.S.A.C. Montpellier. Exposé no. 10 (1987).Google Scholar
  14. A. Choukairi-Dini: B -Convergence des suites adaptées et ensembles de Radon-Nikodym (à paraître) C.R.A.S. Paris, 1994.Google Scholar
  15. J.P. Daures: Version multivoque du théorème de Doob. Ann. Inst. H. Poincaré 9, no. 2, 167–176.Google Scholar
  16. A. Derras: Thèse. Montpellier, 1990.Google Scholar
  17. L. Egghe: Strong convergence of pramarts in Banach spaces. Can. J. Math. 33, no. 2, 357–361.https://doi.org/10.4153/CJM-1981-029-8Google Scholar
  18. L. Egghe: Stopping times techniques in Analysis. Cambridge University Press. Cambridge (1984). London Math. Soc. Lecture Notes ser. Vol. 100.Google Scholar
  19. C. Hess: On multivalued martingales whose values may be unbounded: martingale selectors and Mosco-convergence. S.A.C. Montpellier. Expose no. 8 (1989) et Journal of Multivariate. Analysis vol. 39, no. 1, Octobre (1991).https://doi.org/10.1016/0047-259X(91)90012-QGoogle Scholar
  20. F. Hiai, H. Umegaki: Integrales, conditional expectations and martingales of multivalued functions. J. of Multivariate. Analysis 7 (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  21. H.A. Klei, I. Assani: Parties décomposables de L1E. C.R.A.S. Paris. t. 294, Série I.P. (1982) 533–536.Google Scholar
  22. M. Lavie: Fonction caractéristique d’un ensemble alèatoire. Applications a la convergence de martingales et de sommes d’ensembles aléatoires. C.R.A.S. Paris. t. 315. Série I. (1992) 399–402.Google Scholar
  23. D.Q. Luu: On convergence of multivalued asymptotic martingales. S.A.C. Montpellier. Expose no. 5 (1984).Google Scholar
  24. D.Q. Luu: Représentation of multivalued (regular) uniform amarts. S.A.C. Montpellier. Expose no. 9 (1984).Google Scholar
  25. D.Q. Luu: Applications of set-valued Radon-Nikodym theorems to convergence of multivalued L1 -amarts. Math. Scand. 5.4. (1984) 101–113.https://doi.org/10.7146/math.scand.a-12044Google Scholar
  26. A. Millet, L. Sucheston: Convergence of class of amarts indexed by directed sets. Can. J. Math. 31 (1979) 1033–1046.https://doi.org/10.4153/CJM-1979-095-2Google Scholar
  27. J. Neveu: Convergence presque sûre des martingales multivoques. Ann. Inst. H-Poincaré. Sect. B 8.1–7.Google Scholar
  28. M. Slaby: Strong convergence of vector valued pramarts and subpramarts. Probability and Math. Statistics. vol. 5, Fas. 2 (1985).Google Scholar
  29. M. Talagrand: Some structure results for martingales in the limit and pramarts. The annals of probability. vol. 13, no. 4 (1985).https://doi.org/10.1214/aop/1176992804Google Scholar

Suggested citation

A. Choukairi-Dini. “On Almost Sure Convergence of Vector Valued Pramarts and Multivalued Pramarts.” Journal of Convex Analysis 3 (1996), No. 2, 245–254. https://doi.org/10.68381/jca03016

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.