Limits of First Passage Times to Rare Sets in Regenerative Processes

We consider limits of first passage times to indexed families of nested sets in regenerative processes. The sets are exponentially rare, in the sense that the probability that the process reaches an indexed set in a cycle vanishes exponentially fast in the indexing parameter. Under appropriate formulations of this hypothesis, we prove strong laws, iterated logarithm laws and limits in distribution, both for the index of the rarest set reached in a cycle and for the time to reach a set. An interesting feature of the iterated logarithm laws is an asymmetry in the normalizations for the upper and lower limits. Our results apply to (possibly delayed) wide-sense regenerative processes, as well as those with independent cycles. We illustrate our results with queueing examples.