Key renewal theorem ($g\ast U(t)⟶(\mathbb{E}T{)}^{-1}{\int }_0^{\infty }g$)

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.