The remainder in the Renewal Theorem

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.