Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Martingale property of empirical processes

Authors: Sergio Albeverio, Yeneng Sun and Jiang-Lun Wu
Journal: Trans. Amer. Math. Soc. 359 (2007), 517-527
MSC (2000): Primary 60G42, 60G44; Secondary 03H05, 28E05, 60F15
Published electronically: September 19, 2006
MathSciNet review: 2255184
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.

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

  • 1. S. Albeverio, J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics. Academic Press, Orlando, Florida, 1986. MR 0859372 (88f:03061)
  • 2. R. M. Anderson, A nonstandard representation of Brownian motion and Itô-integration. Israel J. Math. 25 (1976), 15-46. MR 0464380 (57:4311)
  • 3. R. F. Bass, Probabilistic techniques in analysis. Springer-Verlag, New York, 1995. MR 1329542 (96e:60001)
  • 4. J. Berger, H. Osswald, Y. N. Sun and J.-L. Wu, On nonstandard product measure spaces. Illinois J. Math. 46 (2002), 319-330. MR 1936091 (2004g:28021)
  • 5. Y. S. Chow and H. Teicher, Probability theory: independence, interchangeability and martingales (3rd edition), Springer-Verlag, New York, 1997. MR 1476912 (98e:60003)
  • 6. K. L. Chung and Z. X. Zhao, From Brownian motion to Schrödinger's equation. Springer-Verlag, New York, 1995. MR 1329992 (96f:60140)
  • 7. C. Dellacherie and P. A. Meyer, Probabilities and potential. North-Holland, Amsterdam, 1988. MR 0939365 (89b:60002)
  • 8. J. L. Doob, Stochastic processes depending on a continuous parameter. Trans. Amer. Math. Soc. 42 (1937), 107-140. MR 1501916
  • 9. J. L. Doob, Stochastic processes. Wiley, New York, 1953. MR 0058896 (15:445b)
  • 10. J. L. Doob, Classical potential theory and its probabilistic counterpart. Springer-Verlag, New York, 1984. MR 0731258 (85k:31001)
  • 11. D. Duffie, Dynamic asset pricing theory (3rd edition), Princeton University Press, Princeton, N.J., 2001.
  • 12. R. Durret, Probability: theory and examples. Wadsworth, Belmont, California, 1991. MR 1068527 (91m:60002)
  • 13. W. Feller, An introduction to probability theory and Its applications (3rd Edition), Wiley, New York, 1968. MR 0228020 (37:3604)
  • 14. H. Föllmer and A. Schied, Stochastic finance: an introduction in discrete time. Walter de Gruyter, New York, 2002. MR 1925197 (2004h:91051)
  • 15. D. N. Hoover and H. J. Keisler, Adapted probability distributions. Trans. Amer. Math. Soc. 286 (1984), 159-201. MR 0756035 (86m:60096)
  • 16. H. J. Keisler, Hyperfinite model theory. Logic Colloquium 1976 (R. O. Gandy and J. M. E. Hyland eds.), North-Holland, Amsterdam, 1977. MR 0491155 (58:10421)
  • 17. H. J. Keisler, Infinitesimals in probability theory. Nonstandard Analysis and its Applications (N. Cutland ed.), Cambridge University Press, New York, 1988, pp. 106-139. MR 0971065
  • 18. H. J. Keisler and Y. N. Sun, A metric on probabilities, and products of Loeb spaces. J. London Math. Soc. 69 (2004), 258--272. MR 2025340 (2005a:60012)
  • 19. P. A. Loeb, Conversion from standard to nonstandard measure spaces and applications to probability theory. Trans. Amer. Math. Soc. 211 (1975), 113-122. MR 0390154 (52:10980)
  • 20. P. A. Loeb, A nonstandard functional approach to Fubini's theorem. Proc. Amer. Math. Soc. 93 (1985), 343-346. MR 0770550 (86f:28026)
  • 21. P. A. Loeb and M. Wolff (eds.), Nonstandard analysis for the working mathematician. Kluwer Academic Publishers, Amsterdam, 2000. MR 1790871 (2001e:03006)
  • 22. E. Perkins, A global intrinsic characterization of Brownian local time. Ann. Probab. 9 (1981), 800-817. MR 0628874 (82k:60158)
  • 23. K. D. Stroyan and J. M. Bayod, Foundations of infinitesimal stochastic analysis. North-Holland, New York, 1986. MR 0849100 (87m:60001)
  • 24. Y. N. Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN. J. Math. Econom. 29 (1998), 419-503. MR 1627287 (99j:28020)
  • 25. Y. N. Sun, The almost equivalence of pairwise and mutual independence and the duality with exchangeability. Probab. Theory Related Fields 112 (1998), 425-456. MR 1660898 (2000m:60002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60G42, 60G44, 03H05, 28E05, 60F15

Retrieve articles in all journals with MSC (2000): 60G42, 60G44, 03H05, 28E05, 60F15

Additional Information

Sergio Albeverio
Affiliation: Institut für Angewandte Mathematik der Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany

Yeneng Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore

Jiang-Lun Wu
Affiliation: Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom

Keywords: Essential independence, finite-dimensional distributions, empirical process, exact law of large numbers, Loeb product space, Keisler's Fubini theorem, martingale, submartingale, supermartingale.
Received by editor(s): September 16, 2004
Published electronically: September 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society