Abstract
We present various convergence results for multivalued ergodic theorems in Bochner-Gelfand-Pettis integration.
JEL classification: C01, C02
Mathematics Subject Classification (2010): 28B20
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abid, M.: Un théoreme ergodique pour les processus sous-additifs et sur-stationnaires. C.R. Acad. Sci. Paris, Serie A 287, 149–152 (1978)
Akhiat, F., Castaing, C., Ezzaki, F.: Some various convergence results for multivalued martingales. In: Kusuoka, S., Maruyama, T. (eds.) Adv. Math. Econ. 13, 1–33 (2010)
Beer, G.: Support and distance functionals for convex sets. Numer. Funct. Anal. Optim. 10(1–2), 15–36 (1989)
Beer, G.: The slice topology: a viable alternative to Mosco convergence in nonreflexive spaces. Nonlinear Anal. 19(3), 271–290 (1992)
Castaing, C.: Compacité et inf-equicontinuity dans certains espaces de K\(\ddot{o}\)the-Orlicz. Séminaire d’Analyse Convexe, Montpellier, Exposé No 6 (1979)
Castaing, C.: Some various convergence results for normal integrands. Adv. Math. Econ. 15, 1–26 (2011)
Castaing, C., Ezzaki, F.: Variational inequalities for integrand martingales and additive random sequences. Séminaire d’Analyse Convexe, Montpellier, Exposé No 1 (1992) and Acta Mathematica Vietnamica 18(1), 137–171 (1993)
Castaing, C., Ezzaki, F., Hess, C.: Convergence for conditional expectation for unbounded closed convex random sets. Studia Mathematica 124(2), 133–148 (1997)
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, Vol 92. Institute of Mathematics, Polish Academy of Sciences, Warsawa (2011)
Castaing, C., Ezzaki, F., Tahri, K.: Convergences of multivalued pramarts. Journal of Nonlinear and Convex Analysis 11(2), 243–266 (2010)
Castaing, C., Guessous, M.: Convergences in L X 1(μ). Adv. Math. Econ. 1, 17–37 (1999)
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)
Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer Academic Publishers, Dordrecht (2004)
Castaing, C., Saadoune, M.: Dunford-Pettis types theorem and convergences in set-valued integration. Journal of Nonlinear and Convex Analysis 1(1), 37–71 (2000)
Castaing, C., Saadoune, M.: Convergences in a dual space with applications to Fatou lemma. In: Kusuoka, S., Maruyama, T. (eds.) Adv. Math. Econ. 12, pp. 23–69 (2009)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580. Springer-Verlag, Berlin and New York (1977)
Choirat, C., Hess, C., Seri, R.: A functional version of the Birkhoff ergodic theorem for a normal integrand: a variational approach. The Annals of Probability 31, 63–92 (2003)
Dudley, R. M.: Real Analysis And Probability, Chapman-Hall, Mathematics Series. Wadsworth, Inc. (1989)
Fitzpatrick, S., Lewis, A.S.: Weak-star convergence of convex sets. Journal of Convex Analysis 13(3 + 4), 711–719 (2006)
Hiai, F., Umegaki, H.: Integrals, conditional expectations and martingales of multivalued functions. J. Multi. Anal. 7, 149–182 (1977)
Garling, D. J. H.: Subsequence principles for vector-valued random variables. Math. Proc. Cambridge Philos. Soc. 86, 301–311 (1979)
Ghoussoub, N., Steele, M.: Vector valued subadditive process and applications in probability. Ann. Probability 8(1), 83–95 (1990)
Guessous, M.: An elementary proof of Komlos-Revesz theorem in Hilbert space. Journal of Convex Analysis 4, 321–332 (1997)
Jalby, V.: Semi-continuité, convergence et approximation des applications vectorielles. Loi des grands nombres, Université Montpellier II, Laboratoire d’Analyse Convexe, 34095 Montpellier Cedex 05, France, Janvier (1992)
Korf, L. A., Wets, R. J. B.: Random l.s.c functions: an ergodic theorem. Mathematics of Operations Research 26(2), 421–445 (2001)
Krengel, U.: Ergodic Theorems. De Gruyter Studies in Mathematics 6, Berlin, New York (1985)
Krupa, G.: Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces. Studia Mathematica 131(3), (1998)
Mourier, E.: Eléments aléatoires dans un espace de Banach. Ann. Ins. H. Poincare, Sect. B 13, 161–244 (1953)
Neveu, N.: Martingales Etemps discret, Masson et Cie, Editeur (1972)
Peressini, A. L.: Ordered Topological Vector Spaces, Harper’s Series in Modern Mathematics. Harper and Row Publ., New-York (1967)
Schaefer, H.H.: Banach Lattices and Positive Operators, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, 215. Springer-Verlag, Berlin
Schurger, K.: Ergodic theorems for subadditive superstationary families of convex random sets. Z. Wahrsch. verw. Gebiete 62, 125–135 (1983)
Valadier, M.: On conditional expectation of random sets. Annali Math. Pura Appl. 126, 81–91 (1980)
Valadier, M.: Conditional expectation and ergodic theorem for a positive integrand. J. Nonlinear Convex Anal. 1, 233–244 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Japan
About this chapter
Cite this chapter
Castaing, C., Lavie, M. (2012). Some applications of Birkhoff-Kingman ergodic theorem. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 16. Advances in Mathematical Economics, vol 16. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54114-1_1
Download citation
DOI: https://doi.org/10.1007/978-4-431-54114-1_1
Received:
Revised:
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54113-4
Online ISBN: 978-4-431-54114-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)