Renewal theory

Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite expectation. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times.

A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function ${\displaystyle m(t)}$ (expected number of arrivals) and reward function ${\displaystyle g(t)}$ (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of ${\displaystyle m'(t)}$ with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov renewal processes.

Applications include calculating the best strategy for replacing worn-out machinery in a factory; comparing the long-term benefits of different insurance policies; and modelling the transmission of infectious disease, where "One of the most widely adopted means of inference of the reproduction number is via the renewal equation".^[1] The inspection paradox relates to the fact that observing a renewal interval at time t gives an interval with average value larger than that of an average renewal interval.