Key renewal theorem
= Key renewal theorem
{title2=$g*U(t)\longrightarrow(\mathbb ET)^{-1}\int_0^\infty g$}
For positive nonarithmetic interarrival times with law $K$ and mean $m\in(0,\infty]$, let $U=\sum_{j\geq0}K^{*j}$ 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 $z=g+z*K$ has this limit through its <renewal representation>. Arithmetic laws require the corresponding lattice version.