Abstract

We prove results concerning the representation of linear functionals as integrals of a given random quantity X. The existence of such representation is related to the notion of conglomerability, originally introduced by de Finetti and Dubins. We show that this property has interesting applications in probability and in analysis. These include a version of Skorohod theorem, a proof that Brownian motion assumes whatever family of finite dimensional distributions upon a change of the probability measure and a version of the extremal representation theorem of Choquet.

Author information

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

Gianluca Cassese

Dept. of Economics, Statistics and Management, Università Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

gianluca.cassese@unimib.it

Keywords

  • Choquet integral representation
  • conglomerability
  • Riesz representation
  • Skhorohod representation
  • vector lattice

Mathematics Subject Classification

28A25; 46A22, 52A41

References

  1. C. D. Aliprantis, O. Burkinshaw: Positive Operators, Springer Publ., Dordrecht (2006).https://doi.org/10.1007/978-1-4020-5008-4Google Scholar
  2. P. Berti, L. Pratelli, P. Rigo: Asymptotic behaviour of the empirical process for exchangeable data, Stoch. Proc. Appl. 116 (2006) 337–344.https://doi.org/10.1016/j.spa.2005.09.001Google Scholar
  3. P. Berti, L. Pratelli, P. Rigo: Skorohod representation on a given probability space, Probab. Theory Relat. Fields 137 (2007) 277–288.https://doi.org/10.1007/s00440-006-0018-1Google Scholar
  4. K. P. S. Bhaskara Rao, M. Bhaskara Rao: Theory of Charges, Academic Press, London (1993).Google Scholar
  5. R. D. Bourgin: Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lect. Notes Math. 993, Springer-Verlag, Berlin (1983).https://doi.org/10.1007/BFb0069321Google Scholar
  6. G. Cassese: Sure wins, separating probabilities and the representation of linear functionals, J. Math. Anal. Appl. 354 (2009) 558–563.https://doi.org/10.1016/j.jmaa.2009.01.013Google Scholar
  7. G. Choquet (1956): Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes, Sém. Bourbaki 139 (1956) 33–47.Google Scholar
  8. G. Choquet: Determination and study of positive forms on spaces of functions: I, J. Approx. Th. 7 (1973) 325–333.https://doi.org/10.1016/0021-9045(73)90035-XGoogle Scholar
  9. G. Choquet: Determination and study of positive forms on spaces of functions: II, J. Approx. Th. 10 (1974) 358–378.https://doi.org/10.1016/0021-9045(74)90106-3Google Scholar
  10. G. Choquet: Représentation intégrale, in: Measure Theory and its Applications, Lect. Notes Math. 1033 (1983) 114–143.https://doi.org/10.1007/BFb0099850Google Scholar
  11. B. de Finetti: Sulla proprietà conglomerativa delle probabilità subordinate, Rend. R. Istit. Lomb. Sc. Lett. 63 (1930) 3–7.Google Scholar
  12. P. Diaconis, D. Freedman: On the consistency of Bayes estimates, Ann. Probab. 14 (1986) 1–26.https://doi.org/10.1214/aos/1176349830Google Scholar
  13. J. L. Doob: Application of the theory of martingales, Actes Coll. Int. CNRS, Paris (1948) 23–27.Google Scholar
  14. L. E. Dubins, L. J. Savage: How to Gamble if You Must, McGraw-Hill, New York (1965).Google Scholar
  15. L. E. Dubins: Finitely additive conditional probability, conglomerability and disintegrations, Ann. Probab. 3 (1975) 89–99.https://doi.org/10.1214/aop/1176996451Google Scholar
  16. L. E. Dubins: Paths of Finitely Additive Brownian Motion Need not be Bizarre, in: J. Azema, M. Eméry, M. Ledoux, M. Yor (eds.), Séminaire de Probabilités 32, Lect. Notes Math. 1709, Springer-Verlag, Berlin (1999).https://doi.org/10.1007/BFb0096529Google Scholar
  17. N. Dunford, J. T. Schwartz: Linear Operators. Part I, Wiley, New York (1988).Google Scholar
  18. G. A. Edgar: A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 (1975) 354–358.https://doi.org/10.1090/S0002-9939-1975-0372586-2Google Scholar
  19. G. A. Edgar: Extremal integral representations, J. Func. Anal. 23 (1976) 145–161.https://doi.org/10.1016/0022-1236(76)90072-0Google Scholar
  20. B. M. Hill, D. Lane: Conglomerability and countable additivity, in: Bayesian Inference and Decision Techniques. Essays in Honor of Bruno de Finetti, P. K. Goel, A. Zellner (eds.), North-Holland, Amsterdam (1986).Google Scholar
  21. R. Karandikar (1982), A general principle for limit theorems in finitely additive probability, Trans. Amer. Math. Soc. 273 (1982) 541–550.https://doi.org/10.2307/1999927Google Scholar
  22. R. Karandikar: A general principle for limit theorems in finitely additive probability: the dependent case, J. Multivariate Anal. 24 (1988) 189–206.https://doi.org/10.1016/0047-259X(88)90035-8Google Scholar
  23. J. Lukeš, J. Malý, I. Netuka, J. Spurný: Integral Representation Theory. Applications to Convexity, Banach Spaces and Potential Theory, de Gruyter, Berlin (2010).https://doi.org/10.1515/9783110203219Google Scholar
  24. R. R. Phelps: Lectures on Choquet’s Theorem, Lect. Notes Math. 1757, Springer-Verlag, Berlin-Heidelberg (2000).Google Scholar
  25. H. P. Rosenthal: On the Choquet Representation Theorem, in: Functional Analysis, E. W. Odell Jr., H. P. Rosenthal (eds.), Lect. Notes Math. 1332, Springer-Verlag, Berlin-Heidelberg (1988).https://doi.org/10.1007/BFb0081608Google Scholar
  26. M. J. Schervish, T. Seidenfeld, J. B. Kadane: The extent of non-conglomerability of finitely additive probabilities, Z. Wahrsch. verw. Geb. (1984) 205–226.https://doi.org/10.1007/BF00531529Google Scholar
  27. A. Zame: Conglomerability and finite partitions, Proc. Amer. Math. Soc. 102 (1988) 165–168.https://doi.org/10.1090/S0002-9939-1988-0915737-XGoogle Scholar

Suggested citation

G. Cassese. “Conglomerability and Representations.” Journal of Convex Analysis 25 (2018), No. 3, 789–815. https://doi.org/10.68381/jca25055

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.