Asymptotics of the Sample Renewal Function

Let F denote the distribution function of a nonnegative population. Let H denote the corresponding renewal function. Given a random sample of size n from F, the sample renewal functionĤ is defined as the renewal function of the sample distribution function. This is a nonlinear function of the sample distribution function. We give a proof of weak convergence of √n (Ĥ − H) in the Skorokhod topology. This strengthens a results of Frees [Ann. Statist.14 (1986), 1366-1378; Naval Res. Logist.33 (1986), 361-372], who proved asymptotic normality of Ĥ(t) for each fixed t. Grubel and Pitts [Ann. Statist.21 (1993), 1431-1451] proved a more general result by a different method.