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Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test |
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| Authors: | Bischoff Wolfgang Hashorva Enkelejd Hüsler Jürg Miller Frank |
| Affiliation: | 1.Institut für Mathematische Stochastik, Fakult?t für Mathematik, Universit?t Karlsruhe, D-76128, Karlsruhe, Germany ;2.Institut für Mathematische Statistik und Versicherungslehre, Universit?t Bern, Sidlerstr. 5, CH-3012, Bern, Switzerland ; |
| 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. Supported by the Swiss National Science Foundation Grant 20-55586.98. |
| Keywords: | Brownian bridge with trend boundary crossing probability exact asymptotics extreme values large deviations Kolmogorov test |
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