Second-Order Renewal Theorem in the Finite-Means Case

Let F be a distribution function (d.f.) on $(0, \infty )$ and let~U be the renewal function associated with F. If F has a finite first moment~$\mu$, then it is well known that $U(t)$ asymptotically equals $t/\mu$. It is also well known that $U(t)-t/\mu $ asymptotically behaves as $S(t)/\mu, $ where~S denotes the integral of the integrated tail distribution~$F_1$ of~F. In this paper we discuss the rate of convergence of $U(t)-t/\mu -S(t)/\mu $ for a large class of distribution functions. The estimate improves earlier results of Geluk, Teugels, and Embrechts and Omey.