The Asymptotic Validity of Sequential Stopping Rules for Stochastic Simulations

We establish general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters. The asymptotic validity occurs as the prescribed volume of the confidence set approaches 0. There are two requirements: a functional central limit theorem for the estimation process and strong consistency (with-probability-1 convergence) for the variance or "scaling matrix" estimator. Applications are given for: sample means of i.i.d. random variables and random vectors, nonlinear functions of such sample means, jackknifing, Kiefer-Wolfowitz and Robbins-Monro stochastic approximation and both regenerative and nonregenerative steady-state simulation.