Remainder Term Estimates of the Renewal Function

Abstract

Let $\mu$ be a probability measure and $H(x) = \sum^\infty_{n=0} \mu^{n_\ast}(-\infty, x\rbrack$ its renewal function. It is well-known that $H(x) - x/\mu_1 - \mu_2/2\mu^2_1 \rightarrow 0$ as $x \rightarrow +\infty$ if $\mu_1 > 0$ and $\mu$ is a nonlattice measure. ($\mu_k$ is the $k$th moment of $\mu$.) The rate of this convergence is studied under further conditions on $\mu$.