Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Randomized UMD Banach spaces and decoupling inequalities for stochastic integrals

Author: Mark C. Veraar
Journal: Proc. Amer. Math. Soc. 135 (2007), 1477-1486
MSC (2000): Primary 60H05; Secondary 46B09, 46B20
Published electronically: November 14, 2006
MathSciNet review: 2276657
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove the equivalence of decoupling inequalities for stochastic integrals and one-sided randomized versions of the UMD property of a Banach space as introduced by Garling.

References [Enhancements On Off] (What's this?)

  • 1. D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), no. 6, 997-1011. MR 0632972 (83f:60070)
  • 2. D. L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. I, 233-269, North-Holland, Amsterdam, 2001. MR 1863694 (2003b:46009)
  • 3. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 4. D. J. H. Garling, Brownian motion and UMD-spaces, Probability and Banach Spaces (Zaragoza, 1985), 36-49, Lecture Notes in Math. 1221, Springer-Verlag, Berlin, 1986.MR 0875006 (88h:60008)
  • 5. D. J. H. Garling, Random martingale transform inequalities, Probability in Banach spaces VI (Sandbjerg, 1986), 101-119, Progr. Probab., Vol. 20, Birkhäuser, Boston, 1990.MR 1056706 (92a:60111)
  • 6. S. Geiss, A counterexample concerning the relation between decoupling constants and UMD-constants, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1355-1375.MR 1458301 (99f:60011)
  • 7. P. Hitczenko, On tangent sequences of UMD-space valued random vectors, unpublished manuscript, Warsaw, 1988.
  • 8. O. Kallenberg, Foundations of Modern Probability, Second edition, Probability and its Applications, Springer-Verlag, New York, 2002. MR 1876169 (2002m:60002)
  • 9. S. Kwapien and W. A. Woyczynski, Random Series and Stochastic Integrals: Single and Multiple, Probability and its Applications, Birkhäuser, Inc., Boston, 1992. MR 1167198 (94k:60074)
  • 10. M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 23, Springer-Verlag, Berlin, 1991. MR 1102015 (93c:60001)
  • 11. B. Maurey, Système de Haar, Séminaire Maurey-Schwartz 1974-1975: Espaces $ L^{p}$, Applications Radonifiantes et Géométrie des Espaces de Banach, Exp. Nos. I et II, 26 pp. MR 0420839 (54:8851)
  • 12. B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45-90. MR 0443015 (56:1388)
  • 13. T. R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10 (1989), 283-295. MR 1057936 (91i:60010)
  • 14. S. Montgomery-Smith, Concrete representation of martingales, Electron. J. Probab. 3 (1998), No. 15, 15 pp.MR 1658686 (99k:60116)
  • 15. J.M.A.M. van Neerven, M.C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, submitted.
  • 16. J.M.A.M. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2005), 131-170.MR 2109586 (2005j:60107)
  • 17. V. H. de la Peña and E. Giné, Decoupling. From Dependence to Independence Randomly Stopped Processes. $ U$-statistics and Processes. Martingales and Beyond, Probability and its Applications, Springer-Verlag, New York, 1999. MR 1666908 (99k:60044)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60H05, 46B09, 46B20

Retrieve articles in all journals with MSC (2000): 60H05, 46B09, 46B20

Additional Information

Mark C. Veraar
Affiliation: Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands

Keywords: Stochastic integration in Banach spaces, randomized UMD spaces, decoupling inequalities, tangent sequences
Received by editor(s): September 5, 2005
Received by editor(s) in revised form: December 20, 2005
Published electronically: November 14, 2006
Additional Notes: The author is supported by the Netherlands Organisation for Scientific Research (NWO) 639.032.201 and by the Research Training Network HPRN-CT-2002-00281
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society