The Poisson process is the continuous-time object built from the exponential distribution's memorylessness the way the random walk was built from a fixed coin flip. There are three equivalent ways to define it — as a counting process satisfying certain axioms, as constructed directly from i.i.d. exponential inter-arrival times, and as a limit of a binomial approximation — and this note gives all three and proves they agree. Understanding all three matters because different applications reach for different characterisations: the axioms are what a modeller checks against data, the construction is what a simulator implements, and the binomial limit is what explains why Poisson arrivals show up so often in practice (any large number of independent, individually-rare events, summed, looks Poisson).
Learning objectives. By the end of this note you should be able to:
- state the axiomatic definition of a Poisson process and the construction from i.i.d. exponentials;
- prove that the construction satisfies the axioms;
- state and use the binomial-to-Poisson limit theorem, including the case of a general (non-i.i.d.) sum of rare indicators;
- define the non-homogeneous Poisson process as a first generalisation.
In this lesson
- The axiomatic definition
- Construction from i.i.d. exponentials
- The binomial-to-Poisson limit
- A first generalisation: non-homogeneous Poisson processes
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