Renewal theory is one of the main tools in the elementary theory of probability. A renewal process, by itself, does not have a rich enough structure to be of much interest. The main importance of renewal theory is, therefore, largely due to its applications to regenerative processes and to the elegant formalism of the renewal equation. In this light, the importance of the renewal function itself is due to its use as a potential operator and not to its connection with the expectations of the numbers of renewals.
