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Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test

  • Brownian Motion
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Abstract

We consider a boundary crossing probability of a Brownian bridgeB 0 and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB 0 (t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values.

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Authors and Affiliations

  1. Institut für Mathematische Stochastik, Fakultät für Mathematik, Universität Karlsruhe, D-76128, Karlsruhe, Germany

    Wolfgang Bischoff & Frank Miller

  2. Institut für Mathematische Statistik und Versicherungslehre, Universität Bern, Sidlerstr. 5, CH-3012, Bern, Switzerland

    Enkelejd Hashorva & Jürg Hüsler

Authors
  1. Wolfgang Bischoff
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  2. Enkelejd Hashorva
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  3. Jürg Hüsler
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  4. Frank Miller
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Supported by the Swiss National Science Foundation Grant 20-55586.98.

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Bischoff, W., Hashorva, E., Hüsler, J. et al. Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test. Ann Inst Stat Math 55, 849–864 (2003). https://doi.org/10.1007/BF02523397

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  • DOI: https://doi.org/10.1007/BF02523397

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