Skorohod’s representation theorem for sets of probabilities
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- by Martin Dumav and Maxwell B. Stinchcombe PDF
- Proc. Amer. Math. Soc. 144 (2016), 3123-3133 Request permission
Abstract:
We characterize sets of probabilities, $\boldsymbol {\Pi }$, on a measure space $(\Omega ,\mathcal {F})$, with the following representation property: for every measurable set of Borel probabilities, $A$, on a complete separable metric space, $(M,d)$, there exists a measurable $X:\Omega \rightarrow M$ with $A = \{X(P): P \in \boldsymbol {\Pi }\}$. If $\boldsymbol {\Pi }$ has this representation property, then: if $K_n \rightarrow K_0$ is a sequence of compact sets of probabilities on $M$, there exists a sequence of measurable functions, $X_n:\Omega \rightarrow M$ such that $X_n(\boldsymbol {\Pi }) \equiv K_n$ and for all $P \in \boldsymbol {\Pi }$, $P(\{\omega : X_n(\omega ) \rightarrow X_0(\omega )\}) = 1$; if the $K_n$ are convex as well as compact, there exists a jointly measurable $(K,\omega ) \mapsto H(K,\omega )$ such that $H(K_n,\boldsymbol {\Pi }) \equiv K_n$ and for all $P \in \boldsymbol {\Pi }$, $P(\{\omega : H(K_n,\omega ) \rightarrow H(K_0,\omega )\}) = 1$.References
- Gerald Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1269778, DOI 10.1007/978-94-015-8149-3
- David Blackwell and Lester E. Dubins, An extension of Skorohod’s almost sure representation theorem, Proc. Amer. Math. Soc. 89 (1983), no. 4, 691–692. MR 718998, DOI 10.1090/S0002-9939-1983-0718998-0
- Leo Breiman, Lucien Le Cam, and Lorraine Schwartz, Consistent estimates and zero-one sets, Ann. Math. Statist. 35 (1964), 157–161. MR 161413, DOI 10.1214/aoms/1177703737
- Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
- R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358, DOI 10.1017/CBO9780511755347
- Xavier Fernique, Un modèle presque sûr pour la convergence en loi, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 7, 335–338 (French, with English summary). MR 934613
- Itzhak Gilboa, Theory of decision under uncertainty, Econometric Society Monographs, vol. 45, Cambridge University Press, Cambridge, 2009. MR 2493167, DOI 10.1017/CBO9780511840203
- S. Mrówka, On the convergence of nets of sets, Fund. Math. 45 (1958), 237–246. MR 98359, DOI 10.4064/fm-45-1-247-253
- Leonard J. Savage, The foundations of statistics, Second revised edition, Dover Publications, Inc., New York, 1972. MR 0348870
- A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. 1 (1956), 289–319 (Russian, with English summary). MR 0084897
- Pedro Terán, A continuity theorem for cores of random closed sets, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4417–4425. MR 2431058, DOI 10.1090/S0002-9939-08-09412-4
- John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, N.J., 1944. MR 0011937
Additional Information
- Martin Dumav
- Affiliation: Department of Economics, Universidad Carlos III de Madrid, Av. de la Universidad, 30, 28911 Leganés, Madrid, Spain
- Email: mdumav@gmail.com
- Maxwell B. Stinchcombe
- Affiliation: Department of Economics, University of Texas, Austin, Texas 78712-0301
- MR Author ID: 261772
- Email: max.stinchcombe@gmail.com
- Received by editor(s): May 18, 2012
- Received by editor(s) in revised form: July 16, 2014, January 27, 2015, and August 19, 2015
- Published electronically: November 20, 2015
- Communicated by: David Asher Levin
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3123-3133
- MSC (2010): Primary 60B10, 60F99, 91B06
- DOI: https://doi.org/10.1090/proc/12932
- MathSciNet review: 3487242