A Joint Central Limit Theorem for the Sample Mean and Regenerative Variance Estimator

P. W. Glynn and D. L. Iglehart

 Annals of Operations Research, Vol. 8,  41-55 (1987)

Let {V(k): k ≥ 1} be a sequence of independent, identically distributed random vectors in Rd with mean vector μ. The mapping g is a twice differentiable mapping from Rd to R1. Set r = g(μ). A bivariate central limit theorem is proved involving a point estimator for r and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the "return state" with the smallest expected cycle length.