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.