State-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits

Abstract

We study regenerative processes time-changed by state-dependent inverse subordinators. The construction assigns possibly different independent subordinators to measurable classes of excursions and builds a random clock from the corresponding occupation times. Although the resulting process is generally not regenerative, we prove that its transformed excursion point process is again Poisson, using an excursion-wise marking-and-mapping procedure. We apply this to study occupation-time asymptotics. Under regular variation assumptions on the transformed excursion lifetime tails, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling--Kac type limits.