Excursion-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits
Abstract
We study an excursion-dependent generalization of inverse-subordinator time changes for recurrent regenerative processes. Subordinators from an independent family are assigned to measurable classes of excursions and to the distinguished state at which the process is regenerative, and the random clock is constructed from the corresponding occupation times. Although the time-changed process is not necessarily regenerative or Markovian, we show that its transformed excursion point process is again Poisson and is obtained from the original excursion point process by a marking-and-mapping procedure. We also identify the transformed stagnancy component, on the original local-time scale, as a subordinator independent of this point process. We use this representation to study occupation- and waiting-time asymptotics. Under regular variation assumptions on the tails of the Lévy measures of the cumulative excursion lifetimes and stagnancy components, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling--Kac-type limits. We further obtain joint scaling limits for the last and next visit times to the distinguished state.
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BibTeX
@misc{yamato2026excursiondependent,
title = {{Excursion-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits}},
author = {Kosuke Yamato},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.23659},
eprint = {2605.23659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23659},
}
RIS
TY - UNPB
TI - Excursion-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits
AU - Kosuke Yamato
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2605.23659
UR - https://arxiv.org/abs/2605.23659
AB - We study an excursion-dependent generalization of inverse-subordinator time changes for recurrent regenerative processes. Subordinators from an independent family are assigned to measurable classes of excursions and to the distinguished state at which the process is regenerative, and the random clock is constructed from the corresponding occupation times. Although the time-changed process is not necessarily regenerative or Markovian, we show that its transformed excursion point process is again Poisson and is obtained from the original excursion point process by a marking-and-mapping procedure. We also identify the transformed stagnancy component, on the original local-time scale, as a subordinator independent of this point process. We use this representation to study occupation- and waiting-time asymptotics. Under regular variation assumptions on the tails of the Lévy measures of the cumulative excursion lifetimes and stagnancy components, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling--Kac-type limits. We further obtain joint scaling limits for the last and next visit times to the distinguished state.
ER -