Chaos. 2026 Apr 1;36(4):043133. doi: 10.1063/5.0324747.
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 at 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 ϕ(τ) in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both ordinary and equilibrium renewal processes, deriving results for various forms of ϕ(τ). Using saddle-point approximations, we analyze cases where ϕ(τ) follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near τ=0. Additionally, we show how the short-time properties of ϕ(τ) shape the decay of the number of renewals in equilibrium compared to ordinary 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.
PMID:42030067 | DOI:10.1063/5.0324747
