Replicating a renewal process at random times

TL;DR

This paper studies a renewal process subjected to stochastic resetting, revealing diverse growth behaviors of renewal events depending on the interplay of power-law distributions, and introduces a 'dressed' renewal process for Poissonian resetting.

Contribution

It provides a detailed analysis of the statistical properties of renewal processes with random resets, including the derivation of a 'dressed' renewal process under Poissonian resetting, and explores implications for first passage and random walks.

Findings

Total renewal events can grow linearly or subextensively depending on distribution exponents.

The most regular process dominates the behavior across the phase diagram.

A 'dressed' renewal process describes the statistics under Poissonian resetting.

Abstract

We replicate a renewal process at random times, which is equivalent to nesting two renewal processes, or considering a renewal process subject to stochastic resetting. We investigate the consequences on the statistical properties of the model of the intricate interplay between the two probability laws governing the distribution of time intervals between renewals, on the one hand, and of time intervals between resettings, on the other hand. In particular, the total number ${\mathcal N}_t$ of renewal events occurring within a specified observation time exhibits a remarkable range of behaviours, depending on the exponents characterising the power-law decays of the two probability distributions. Specifically, ${\mathcal N}_t$ can either grow linearly in time and have relatively negligible fluctuations, or grow subextensively over time while continuing to fluctuate. These behaviours highlight the dominance of the most regular process across all regions of the phase diagram. In the presence of Poissonian resetting, the statistics of ${\mathcal N}_t$ is described by a unique `dressed' renewal process, which is a deformation of the renewal process without resetting. We also discuss the relevance of the present study to first passage under restart and to continuous time random walks subject to stochastic resetting.