Continuous Ocone Martingales as Weak Limits of Rescaled Martingales

Harry van Zanten (Vrije Universiteit Amsterdam)

Abstract


Consider a martingale $M$ with bounded jumps and two sequences $a_n, b_n \to \infty$. We show that if the rescaled martingales $$ M^n_t =\frac{1}{\sqrt{a_n}}M_{b_n t}$$ converge weakly, then the limit is necessarily a continous Ocone martingale. Necessary and sufficient conditions for the weak convergence of the rescaled martingales are also given.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 215-222

Publication Date: November 28, 2002

DOI: 10.1214/ECP.v7-1062

Supplementary Files

Correction to ``Continuous Ocone Martingales as Weak Limits of Rescaled Martingales" (pdf file) (34KB)
Correction to ``Continuous Ocone Martingales as Weak Limits of Rescaled Martingales" (ps file) (114KB)


References

  1. Beghdadi-Sakrani, S. (2002). The uniqueness class of continuous local martingales. Bernoulli 8(2), 207--217. Math. Review
  2. Coquet, F., Mémin, J. and Vostrikova, L. (1994). Rate of convergence in the functional limit theorem for likelihood ratio processes. Math. Methods Statist. 3(2), 89--113. Math. Review
  3. Dubins, L.E., Émery, M. and Yor, M. (1993). On the Lévy transformation of Brownian motions and continuous martingales. In Séminaire de Probabilités, XXVII, pp. 122--132. Springer, Berlin. Math. Review
  4. Jacod, J. and Shiryaev, A.N. (1987). Limit theorems for stochastic processes. Springer-Verlag, Berlin. Math. Review
  5. Karatzas, I. and Shreve, S.E. (1991). Brownian motion and stochastic calculus. Springer-Verlag, New York. Math. Review
  6. Kubilius, K. (1985). The rate of convergence in the functional central limit theorem for semimartingales. Litovsk. Mat. Sb. 25(1), 84--96. Math. Review
  7. Liptser, R.S. and Shiryayev, A.N. (1989). Theory of martingales. Kluwer Academic Publishers Group, Dordrecht. Math. Review
  8. Monroe, I. (1972). On embedding right continuous martingales in {B}rownian motion. Ann. Math. Statist. 43, 1293--1311. Math. Review
  9. Ocone, D.L. (1993). A symmetry characterization of conditionally independent increment martingales. In Barcelona Seminar on Stochastic Analysis, 1991, pp. 147--167. Birkh"auser, Basel. Math. Review
  10. Revuz, D. and Yor, M. (2001). Continuous martingales and Brownian motion. Springer-Verlag, Berlin, third edition. Math. Review
  11. Van Zanten, J.H. (2000). A multivariate central limit theorem for continuous local martingales. Statist. Probab. Lett. 50(3), 299--235. Math. Review
  12. Vostrikova, L. and Yor, M. (2000). Some invariance properties (of the laws) of Ocone's martingales. In Séminaire de Probabilités, XXXIV, pp. 417--431. Springer, Berlin. Math. Review
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.