Renewal theory with fat tailed distributed sojourn times: typical versus rare

TL;DR

This paper explores rare events in renewal processes with heavy-tailed waiting times, introducing new methods beyond traditional large deviation theory to analyze atypical fluctuations.

Contribution

It develops a framework using non-normalized densities to describe rare fluctuations in renewal processes with fat-tailed distributions, extending beyond classical fluctuation laws.

Findings

Non-normalized densities effectively describe rare fluctuations.

Numerical simulations confirm deviations from classical laws.

New tools are proposed for analyzing rare events in heavy-tailed renewal processes.

Abstract

Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modelling and so typical fluctuations of observables of interest have been investigated in detail. To describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered. Here we investigate the large deviations of the number of renewals, the forward and backward recurrence time, the occupation time, and the time interval straddling the observation time. We show how non-normalized densities describe these rare fluctuations, and how moments of certain observables are obtained from these limiting laws. Numerical simulations illustrate our results showing the deviations from arcsine, Dynkin, Darling-Kac, L{\'e}vy and Lamperti laws.