The Poisson process's independent-increments property was shown to rest entirely on the memorylessness of its exponential inter-arrival times. Drop that one assumption — let inter-arrival times be i.i.d. from any distribution, not necessarily exponential — and independent increments is genuinely lost, but a rich and still-tractable theory survives: the renewal process. This note builds the object and its central analytic tool, the renewal equation — an integral equation the expected count must satisfy, which generalises to a technique (the "renewal-type equation") used repeatedly throughout applied probability whenever a quantity can be decomposed by conditioning on the first event.
Learning objectives. By the end of this note you should be able to:
- define a renewal process and identify a Markov chain's return times as a renewal process;
- derive the renewal equation for the renewal function ;
- state the general renewal-type equation and its unique-bounded-solution property;
- verify the renewal function for a Poisson process directly from the renewal equation.
In this lesson
- The renewal process
- The renewal function
- The general renewal-type equation
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