The remainder in the Renewal Theorem

TL;DR

This paper investigates the precise asymptotic behavior of the renewal function's remainder term for renewal processes with heavy-tailed step distributions, extending known results to all cases except a critical index.

Contribution

It provides the exact asymptotic behavior of the renewal function's remainder term for heavy-tailed distributions with index between 1 and 2, excluding the critical case of 3/2.

Findings

Exact asymptotics for the renewal function's remainder term for most tail indices.

Results apply to both renewal processes and random walks.

Special treatment and correction for the critical case of α=3/2.

Abstract

If the step distribution in a renewal process has finite mean and regularly varying tail with index -{\alpha}, 1<{\alpha}<2, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we show that, without making any additional assumptions, it is possible to give, in all cases except for {\alpha}=3/2 , the exact asymptotic behaviour of the next term. In the case {\alpha}=3/2 the result is exact to within a slowly varying correction. Similar results are shown to hold in the random walk case.