Local behaviour of the remainder in Renewal theory

TL;DR

This paper investigates the asymptotic behavior of the renewal mass and density functions for discrete and continuous renewal processes with positive mean and heavy-tailed distributions.

Contribution

It provides new asymptotic estimates for the renewal mass function and density in cases with regularly varying tails, extending renewal theory results.

Findings

Asymptotic estimates for renewal mass function in discrete case

Results for renewal density function in continuous case

Extension of renewal theory to heavy-tailed distributions

Abstract

Several terms in an asynptotic estimate for the renewal mass function ina discrete random walk which has positive mean and regularly varying right-hand tail are given. Similar results are given for the renewal density function in the absolutely continuous case.