 Renewal representation

For positive interarrivals with law $K$, the unique locally bounded solution of $z=g+z\ast K$ is $z=g\ast U={\sum }_{j\ge 0}g\ast K^{\ast j}$, where $U$ is the renewal measure. Iteration proves the identity: the residual is bounded on each compact interval by a constant times the probability that the sum of many positive interarrivals stays in that interval, which tends to zero.