Statistics of a Large Number of Renewals in Equilibrium and Non-Equilibrium Renewal Processes

TL;DR

This paper analyzes the probability distribution of the number of renewals in large renewal processes during short times, highlighting how the form of the waiting time distribution influences behavior in both equilibrium and non-equilibrium systems.

Contribution

It provides new theoretical results on how the specific properties of the waiting time distribution affect renewal statistics in the large-number regime, including rare event analysis.

Findings

Large-number-of-renewals behavior depends on $\, au$ distribution details.

Short-time properties influence decay of renewal counts.

Results aid understanding of rare events in complex systems.

Abstract

The renewal process is a key statistical model for describing a wide range of stochastic systems in Physics. This work investigates the behavior of the probability distribution of the number of renewals in renewal processes in the short-time limit, with a focus on cases where the number of renewals is large. We find that the specific details of the sojourn time distribution $\phi(\tau)$ in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both non-equilibrium and equilibrium renewal processes, deriving results for various forms of $\phi(\tau)$. Using saddle point approximations, we analyze cases where $\phi(\tau)$ follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near $\tau = 0$. Additionally, we show how the short-time properties of $\phi(\tau)$ shape the decay of the number of renewals in equilibrium compared to non-equilibrium renewal processes. The probability of the number of renewals plays a crucial role in determining rare event behaviors, such as Laplace tails. The results obtained here are expected to help advance the development of a theoretical framework for rare events in transport processes in complex systems.