Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend

Abstract

Let us consider a signal-plus-noise model γh(z)+B 0(z), z ∈ [0,1], where γ > 0, h: [0,1] → ℝ, and B 0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for γ→∞, that is P (sup zε [0,1] w(z)(γ h(z)+B 0(z))>c), for γ→∞, (1) where w: [0,1]→ [0,∞ is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H 0: h≡ 0 against the alternative K: h>0 in the signal-plus-noise model.

This is a preview of subscription content, access via your institution.

References

  1. A. Basilevsky, Applied Matrix Algebra in the Statistical Sciences, North-Holland: New York, NY, 1983.

    Google Scholar 

  2. J. Durbin, “The first-passage density of the Brownian motion process to a curved boundary,” J. Appl. Probab. vol. 29 pp. 291–304, 1992.

    Google Scholar 

  3. J. A. Hájek, and Z. Sidák, Theory of Rank Tests, Academic Press: New York, 1967.

    Google Scholar 

  4. E. Hashorva and J. Hüsler, “On asymptotics of multivariate integrals with applications to records,” Stochastic Models vol. 18,no. 1 pp. 41–69, 2002.

    Google Scholar 

  5. A. Janssen and M. Kunz, “Boundary crossing probabilities for piecewise linear boundary functions,” Preprint, 2000.

  6. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer: New York, 1991.

    Google Scholar 

  7. M. Ledoux, Isoperimetry and Gaussian Analysis, Lectures on Probability Theory and Statistics. Lecture Notes in Math., 1648, Springer: New York, 1996.

    Google Scholar 

  8. A. A. Mogul'skii, “Common approach to studying the probability of large and small deviations for random walks,” Probability Theory and Mathematical Statistics, Lecture Notes in Math., 1021, Springer, pp. 452–460, 1983.

  9. A. A. Mogul'skii, “Large deviations of the Wiener process,” Advances in Probability Theory: Limit Theorems and Related Problems, Transl. Ser. Math. Eng., pp. 41–85, 1984.

  10. A. Novikov, V. Frishling, and N. Kordzakhia, “Approximations of boundary crossing probabilities for a Brownian motion,” J. Appl. Prob. vol. 36 pp. 1019–1030, 1999.

    Google Scholar 

  11. T. H. Scheike, “A boundary crossing result for Brownian motion,” J. Appl. Probab. vol. 29 pp. 448–453, 1992.

    Google Scholar 

  12. D. Siegmund, “Boundary crossing probabilities and statistical applications,” Ann. Statist. vol. 14 pp. 361–404, 1986.

    Google Scholar 

  13. S. R. S. Varadhan, Large Deviations and Applications, S.I.A.M.: Philadelphia, 1984.

    Google Scholar 

  14. L. Wang and K. Pötzelberger, “Boundary crossing probability for Brownian motion and general boundaries,” J. Appl. Probab. vol. 34 pp. 54–65, 1997.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bischoff, W., Miller, F., Hashorva, E. et al. Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend. Methodology and Computing in Applied Probability 5, 271–287 (2003). https://doi.org/10.1023/A:1026242019110

Download citation

  • Brownian bridge with trend
  • boundary crossing probability
  • asymptotic results
  • large deviations
  • signal-plus-noise model
  • tests of Kolmogorov type