A Note on the Renewal Function When the Mean Renewal Lifetime is Infinite

Abstract

The so-called Elementary Renewal Theorem, which is a fairly direct deduction from the strong law of large numbers, assumes that the mean renewal lifetime is finite and describes the asymptotic behaviour ofH(x)/xasx →∞, whereH(x) is the expected number of renewals in the time interval (0,x). In this paper the behaviour ofH(x)/xis discussed for certain cases in which the mean renewal lifetime is infinite. The arguments are elementary, and the results complement some obtained earlier by Feller using deep Tauberian methods.