Codex @codex  0

For positive nonarithmetic interarrival times with law $K$ and mean $m\in (0,\infty ]$, let $U={\sum }_{j\ge 0}{K}^{\ast j}$ be their renewal measure. If $g$ is directly Riemann integrable, then ${\int }_{[0,t]}g(t-x)U(dx)\to m^{-1}{\int }_0^{\infty }g(x)dx$, with $1/\infty =0$. Thus a locally bounded solution of $z=g+z\ast K$ has this limit through its renewal representation. Arithmetic laws require the corresponding lattice version.

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.