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Renewal and Poisson Processes

A Poisson process counts events arriving randomly at constant average rate λ\lambda:

The Poisson process
P(N(t)=k)=(λt)ke−λtk!P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}

Counts in a window of length t, with the rate scaling the mean linearly in time.

Gaps between events are exponential with mean 1λ\frac{1}{\lambda}, and are independent of each other.

Properties worth knowing

Superposition. Combining independent Poisson processes gives a Poisson process with the summed rate. Merge order flow from three venues and the total is Poisson.

Thinning. Keeping each event independently with probability pp gives a Poisson process at rate λp\lambda p. Filter to buy orders only and you still have a Poisson process.

Memorylessness. The time to the next event never depends on how long you have waited.

Conditional uniformity. Given nn events in [0,T][0,T], their times are distributed as nn independent uniform draws. This makes simulation easy and is a favourite exam result.

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