Renewal equation

Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. Applications include calculating the best strategy for replacing worn-out machinery in a factory (example below) and comparing the long-term benefits of different insurance policies.

A renewal process is a generalization of the Poisson process. In essence, the Poisson process is a continuous-time Markov process on the positive integers (usually starting at zero) which has independent identically distributed holding times at each integer ${\displaystyle i}$ (exponentially distributed) before advancing (with probability 1) to the next integer:${\displaystyle i+1}$. In the same informal spirit, we may define a renewal process to be the same thing, except that the holding times take on a more general distribution. (Note however that the independence and identical distribution (IID) property of the holding times is retained).

Let ${\displaystyle S_{1},S_{2},S_{3},S_{4},S_{5},\ldots }$ be a sequence of positive independent identically distributed random variables such that

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