The Elementary and Key Renewal Theorems

Solving the renewal equation exactly for $m(t)$ is often intractable beyond the exponential case. This note supplies what is needed in practice instead: the long-run rate at which renewals accumulate ($N(t)/t\to1/\mu$, a strong law of large numbers) and two increasingly refined statements about the renewal function itself — the elementary renewal theorem ($m(t)/t\to1/\mu$) and, stated without proof, the sharper key renewal theorem, the tool this topic's inspection-paradox derivation and renewal-reward capstone both ultimately lean on.

Learning objectives. By the end of this note you should be able to: