Conditional boundary crossing probabilities and two-stage tests for a change-point

For normal random walks S-1,S-2,..., a large deviation approximation is obtained for the probability of the event that the stochastic process {S-n} crosses over the boundary {n(1-n/m)}(1/2) in the time interval m(0) less than or equal to n less than or equal to m(1) (<m) conditionally on S-m0 = S-m1 being fixed. The result is applied to a change-point problem to approximate the significance level of the two-stage test, which is defined as a stochastic convex combination of the modified likelihood ratio test and Pettitt's test.