Some Almost-Sure Convergence Properties Useful in Sequential Analysis

Kim and Nelson propose sequential procedures for selecting the simulated system with the largest steady-state mean from a set of alternatives that yield stationary output processes. Each procedure uses a triangular continuation region so that sampling stops when the relevant test statistic first reaches the region's boundary. In applying the generalized continuous mapping theorem to prove the asymptotic validity of these procedures as the indifference-zone parameter tends to zero, we are given (i) a sequence of functions on the unit interval (which are right-continuous with left-hand limits) converging to a realization of a certain Brownian motion process with drift; and (ii) a sequence of triangular continuation regions corresponding to the functions in sequence (i) and converging to the triangular continuation region for the Brownian motion process. From each function in sequence (i) and its corresponding continuation region in sequence (ii), we obtain the associated boundary-hitting point; and we prove that the resulting sequence of such points converges almost surely to the boundary-hitting point for the Brownian motion process. We also discuss the application of this result to a statistical process-control scheme for autocorrelated data and to other selection procedures for steady-state simulation experiments.