Return-state independent quantities in regenerative simulation

An important parameter of the regenerative method of simulation output analysis is the choice of return state used for blocking observations. Computational experience has shown that the statistical properties of estimators based on different regeneration points can vary widely. This paper studies the limiting joint distribution of the normalized regenerative point and standard-deviation estimators for general state-space Markov chains. The asymptotic covariance between the point and standard-deviation estimators is shown to be the same for all return states, and the quantity is related to the normalized skewness of the partial sums of the chain. Since the covariance is constant, it follows that the choice of return state that minimizes the asymptotic variance of the standard-deviation estimator will maximize the correlation between the point and standard-deviation estimator. Consideration of asymptotic variance and covariance alone suggests that confidence intervals have the best coverage probabilities if the asymptotic variance of the standard-deviation estimator is minimized (thereby maximizing the correlation).