Convergence theorems for adapted sequences of random sets

References

• Van Cutsem, B. (1972). Martingales de convexes fermés aléatoires en dimension finie. Ann. Inst. H. Poincaré Sect. B (N.S) 8:365–385.
   Google Scholar
• Neveu, J. (1972). Convergence presque sûre de martingales multivoques. Ann. Inst. H. Poincaré Sect. B (N.S) 8:1–7.
   Google Scholar
• Daurès, J.-P. (1973). Version multivoque du théorème de Doob. Ann. Inst. H. Poincaré Sect. B (N.S) 9:167–176.
   Google Scholar
• Alò, R. A., de Korvin, A., Roberts, C. (1979). The optional sampling theorem for convex set-valued martingales. J. Reine Angew. Math. 310:1–6.
   Web of Science ®Google Scholar
• Bagchi, S. (1985). On a.s. convergence of classes of multivalued asymptotic martingales. Ann. Inst. H. Poincaré Probab. Statist. 21(4):313–321.
   Web of Science ®Google Scholar
• Hess, C. (1991). On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence. J. Multivariate Anal. 39(1):175–201. DOI: 10.1016/0047-259X(91)90012-Q.
   Web of Science ®Google Scholar
• Hiai, F., Umegaki, H. (1977). Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7(1):149–182. DOI: 10.1016/0047-259X(77)90037-9.
   Web of Science ®Google Scholar
• de Korvin, A., Kleyle, R. A. (1985). Convergence theorem for convex set-valued supermartingales. Stoch Anal. Appl. 3(4):433–445. DOI: 10.1080/07362998508809072.
   Web of Science ®Google Scholar
• Li, S., Ogura, Y. (1998). Convergence of set-valued sub- and supermartingales in the Kuratowski-Mosco sense. Ann. Probab. 26(3):1384–1402. DOI: 10.1214/aop/1022855757.
   Web of Science ®Google Scholar
• Li, S., Ogura, Y. (2003). A convergence theorem of fuzzy-valued martingales in the extended Hausdorff metric H∞. Fuzzy Sets Syst. 135(3):391–399. DOI: 10.1016/S0165-0114(02)00145-8.
   Web of Science ®Google Scholar
• Lu’u, D. Q. (1984). Applications of set-valued Radon-Nikodým theorems to convergence of multivalued L1-amarts. Math. Scand. 54(1):101–113.
   Web of Science ®Google Scholar
• Papageorgiou, N. S. (1987). A convergence theorem for set-valued supermartingales with values in a separable Banach space. Stoch Anal. Appl. 5(4):405–422. DOI: 10.1080/07362998708809128.
   Web of Science ®Google Scholar
• Papageorgiou, N. S. (1991). Convergence and representation theorems for set-valued random processes. Probab. Math. Statist. 11(2):253–269.
   Google Scholar
• Papageorgiou, N. S. (1992). Weak convergence of random sets in Banach spaces. J. Math. Anal. Appl. 164(2):571–589. DOI: 10.1016/0022-247X(92)90136-2.
   Web of Science ®Google Scholar
• Papageorgiou, N. S. (1995). On the conditional expectation and convergence properties of random sets. Trans. Am. Math. Soc. 347(7):2495–2515. DOI: 10.1090/S0002-9947-1995-1290728-9.
   Web of Science ®Google Scholar
• Wang, R. (2001). Essential (convex) closure of a family of random sets and its applications. J. Math. Anal. Appl. 262(2):667–687.
   Web of Science ®Google Scholar
• Wang, Z. P., Xue, X. H. (1994). On convergence and closedness of multivalued martingales. Trans. Am. Math. Soc. 341(2):807–827. DOI: 10.1090/S0002-9947-1994-1154544-X.
   Web of Science ®Google Scholar
• Li, S., Ogura, Y., Kreinovich, V. (2002). Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Dordrecht: Kluwer Academic Publishers Group.
   Google Scholar
• Molchanov, I. (2005). Theory of Random Sets. London: Springer-Verlag.
   Google Scholar
• Hu, S., Papageorgiou, N. S. (1997). Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and Its Applications. Dordrecht: Kluwer Academic Publishers, p. 419.
   Google Scholar
• Egghe, L. (1984). Stopping Time Techniques for Analysts and Probabilists (London Mathematical Society Lecture Note Series, 100). Cambridge: Cambridge University Press.
   Google Scholar
• Dunford, N., Schwartz, J. T. (1988). Linear Operators. Part I. General Theory. New York: John Wiley & Sons.
   Google Scholar
• Chacon, R. V., Sucheston, L. (1975). On convergence of vector-valued asymptotic martingales. Z. Wahrscheinlichkeitstheorie Verw. Gebiete. 33(1):55–59. DOI: 10.1007/BF00539861.
   Google Scholar
• Edgar, G. A., Sucheston, L. (1976). Amarts: a class of asymptotic martingales. A discrete parameter. J. Multivariate Anal. 6(2):193–221. DOI: 10.1016/0047-259X(76)90031-2.
   Web of Science ®Google Scholar
• Gasiński, L., Papageorgiou, N. S. (2014). Exercises in Analysis. Part 1, Problem Books in Mathematics. Cham: Springer.
   Google Scholar
• Costé, A. (1977). Pallu de la barriere, R. Radon-Nikodým theorems for set-valued measures whose values are convex and closed. Comment. Math. Prace Mat. 20:283–309.
   Google Scholar
• Gasiński, L., Papageorgiou, N. S. (2006). Nonlinear Analysis. Series in Mathematical Analysis and Applications, 9. Boca Raton, FL: Chapman & Hall/CRC.
   Google Scholar
• Neveu, J. (1975). Discrete-Parameter Martingales. Amsterdam: North-Holland Publishing Co. (New York: American Elsevier Publishing Co., Inc.)
   Google Scholar
• Gasiński, L., Papageorgiou, N. S. (2016). Exercises in Analysis. Part 2, Problem Books in Mathematics. Cham: Springer.
   Google Scholar
• Uhl, J. J. (1977). Pettis mean convergence of vector-valued asymptotic martingales. Z. Wahrscheinlichkeitstheorie Verw. Gebiete. 37(4):291–295. DOI: 10.1007/BF00533421.
   Google Scholar
• Bellow, A. (1978). Uniform amarts: a class of asymptotic martingales for which strong almost sure convergence obtains. Z Wahrscheinlichkeitstheorie Verw. Gebiete. 41(3):177–191. DOI: 10.1007/BF00534238.
   Google Scholar
• Diestel, J., Uhl, J. J. (1977). Vector Measures. Providence, RI: American Mathematical Society.
   Google Scholar
• Denkowski, Z., Migórski, S., Papageorgiou, N. S. (2003). An Introduction to Nonlinear Analysis: Theory. New York: Kluwer/Plenum.
   Google Scholar
• Brunel, A., Sucheston, L. (1976). Sur les amarts a valeurs vectorielles. C. R. Acad. Sci. Paris Sér. A-B 283(15):A1037–A1040.
   Google Scholar
• Chatterji, S. D. (1976). Vector-valued martingales and their applications, in “Probability in banach spaces.” Lect. Note Math. 526:33–51.
   Google Scholar
• Schaefer, H. H. (1974). Banach Lattices and Positive Pperators. New York; Heidelberg: Springer-Verlag.
   Google Scholar
• Davis, W. J., Ghoussoub, N., Lindenstrauss, J. (1981). A lattice renorming theorem and applications to vector-valued processes. Trans. Am. Math. Soc. 263(2):531–540. DOI: 10.1090/S0002-9947-1981-0594424-2.
   Web of Science ®Google Scholar
• Słaby, M. (1982). Convergence of submartingales and amarts in Banach lattices. Bull. Acad. Polon. Sci. Sér. Sci. Math. 30:291–299.
   Google Scholar
• Ghoussoub, N. (1977). Banach lattices valued amarts. Ann. Inst. H. Poincaré Sect. B (N.S) 13(2):159–169.
   Web of Science ®Google Scholar
• Megginson, R. E. (1998). An Introduction to Banach Space Theory. New York: Springer-Verlag.
   Google Scholar