Chapter 2 — Poisson Processes

1. Counting-process definition

A Poisson process \((N(t))_{t\ge0}\) with rate \(\lambda>0\) starts at zero, has independent increments, stationary increments and

\[N(t)-N(s)\sim\operatorname{Poisson}(\lambda(t-s)), \qquad t>s.\]

Hence

\[\mathbb P(N(t)-N(s)=k)=e^{-\lambda(t-s)}\frac{[\lambda(t-s)]^k}{k!}.\]

2. Arrival times

Let \(T_n\) be the time of the \(n\)-th event. The inter-arrival times are i.i.d.

\[T_n-T_{n-1}\sim\operatorname{Exp}(\lambda).\]

This gives two equivalent constructions of the process: simulate exponential waiting times or simulate counts over intervals.

3. Basic consequences

The increment over length \(h\) has mean and variance

\[\mathbb E[N(t+h)-N(t)]=\lambda h,\]
\[\operatorname{Var}(N(t+h)-N(t))=\lambda h.\]

The equality of mean and variance is characteristic of the Poisson count model.

4. Conditional structure

Conditionally on the total number of arrivals in an interval, event times have the familiar order-statistics interpretation. This is useful for deriving arrival-time probabilities and simulation algorithms.

5. Superposition and thinning

Independent Poisson processes can be superposed to form a Poisson process with rate equal to the sum of the component rates. Conversely, thinning a Poisson process with constant retention probability gives another Poisson process with the scaled rate.

6. Non-homogeneous extension

If the intensity is time-dependent, \(\lambda(t)\), the increment distribution is controlled by the integrated intensity

\[\Lambda(s,t)=\int_s^t\lambda(u)\,du.\]

Then

\[N(t)-N(s)\sim\operatorname{Poisson}(\Lambda(s,t)).\]

7. Simulation

Use an explicit generator in software experiments so that the same mathematical model produces reproducible paths.

8. StochX objects

PoissonProcess represents the homogeneous model and NonHomogeneousPoissonProcess represents the intensity-driven extension. Use them for event/arrival models; use the time-series namespace for regularly indexed measurements.

See the Poisson-process guide and API pages.