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.