Forward-convex convergence in probability of sequences of nonnegative random variables
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- by Constantinos Kardaras and Gordan Žitković PDF
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Abstract:
For a sequence $(f_n)_{n \in \mathbb {N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $(h_n)_{n \in \mathbb {N}}$ with $h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $n \in \mathbb {N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.References
- Klaus Bichteler, Stochastic integrators, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 5, 761–765. MR 537627, DOI 10.1090/S0273-0979-1979-14655-X
- Klaus Bichteler, Stochastic integration and $L^{p}$-theory of semimartingales, Ann. Probab. 9 (1981), no. 1, 49–89. MR 606798
- W. Brannath and W. Schachermayer, A bipolar theorem for $\mathbb {L}^0_+ (\Omega ,\mathcal {F}, \mathbb {P})$, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 349–354. MR 1768009 (2001d:46019)
- A. V. Buhvalov and G. Ja. Lozanovskiĭ, Sets closed in measure in spaces of measurable functions, Dokl. Akad. Nauk SSSR 212 (1973), 1273–1275 (Russian). MR 0346507
- F. Delbaen and W. Schachermayer, A compactness principle for bounded sequences of martingales with applications, Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996) Progr. Probab., vol. 45, Birkhäuser, Basel, 1999, pp. 137–173. MR 1712239
- Freddy Delbaen and Walter Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), no. 3, 463–520. MR 1304434 (95m:90022b)
- C. Dellacherie, Un survol de la théorie de l’intégrale stochastique, Stochastic Process. Appl. 10 (1980), no. 2, 115–144 (French, with English summary). MR 587420, DOI 10.1016/0304-4149(80)90017-4
- Damir Filipović, Michael Kupper, and Nicolas Vogelpoth, Separation and duality in locally $L^0$-convex modules, J. Funct. Anal. 256 (2009), no. 12, 3996–4029. MR 2521918, DOI 10.1016/j.jfa.2008.11.015
- N. J. Kalton, N. T. Peck, and James W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777, DOI 10.1017/CBO9780511662447
- Constantinos Kardaras, Numéraire-invariant preferences in financial modeling, Ann. Appl. Probab. 20 (2010), no. 5, 1697–1728. MR 2724418, DOI 10.1214/09-AAP669
- J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 210177, DOI 10.1007/BF02020976
- D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab. 9 (1999), no. 3, 904–950. MR 1722287, DOI 10.1214/aoap/1029962818
- E. M. Nikišin, A certain problem of Banach, Dokl. Akad. Nauk SSSR 196 (1971), 774–775 (Russian). MR 0277950
- C. Stricker, Une caractérisation des quasimartingales, Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975), Lecture Notes in Math., Vol. 465, Springer, Berlin, 1975, pp. 420–424 (French). MR 0423516
- Gordan Žitković, Convex compactness and its applications, Math. Financ. Econ. 3 (2010), no. 1, 1–12. MR 2651515, DOI 10.1007/s11579-010-0024-z
Additional Information
- Constantinos Kardaras
- Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
- Email: kardaras@bu.edu
- Gordan Žitković
- Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712
- Email: gordanz@math.utexas.edu
- Received by editor(s): October 17, 2010
- Received by editor(s) in revised form: February 3, 2011, and July 16, 2011
- Published electronically: July 9, 2012
- Additional Notes: The authors would like to thank Freddy Delbaen and Ted Odell for valuable help, numerous conversations and shared expertise, as well as the anonymous referee for constructive comments and suggestions
Both authors acknowledge partial support by the National Science Foundation, the first author under award number DMS-0908461, and the second author under award number DMS-0706947. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation. - Communicated by: Richard C. Bradley
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 919-929
- MSC (2010): Primary 46A16, 46E30, 60A10
- DOI: https://doi.org/10.1090/S0002-9939-2012-11373-5
- MathSciNet review: 3003684