Conditional Boundary Crossing Probabilities, with Applications to Change-Point Problems

Abstract

For normal random walks $S_1, S_2,\ldots$, formed from independent identically distributed random variables $X_1, X_2,\ldots$, we determine the asymptotic behavior under regularity conditions of $P(S_n > mg(n/m) \text{for some} n < m\mid S_m = m\xi_0, U_m = m\lambda_0), \quad\xi_0 < g(1),$ where $U_m = X^2_1 + \cdots + X^2_m$. The result is applied to a normal change-point problem to approximate null distributions of test statistics and to obtain approximate confidence sets for the change-point.