On Confidence Intervals for Cyclic Regenerative Processes *

Recently, Sargent and Shanthikumar [5] developed an interesting new variance reduction technique designed to exploit the stochastic structure associated with a cyclic regenerative process. Our purpose here is to study precise conditions under which the confidence intervals proposed in [5] are asymptotically valid. This analysis will provide us with the side benefit of obtaining an optimal way of implementing the variance reduction procedures introduced there. To be precise, we will obtain the minimum variance estimate in the class of estimates proposed in [5]. We will also determine conditions under which the minimum variance estimate achieves a lower variance than that of the standard regenerative method (see Crane and Lemoine [4] for a description of the standard procedure). We will use the convention that assumptions in force throughout the entire paper will be prefixed by A (e.g. Ai) whereas all others will be prefixed by B. We can now state our basic assumptions for the problem: