# The Poisson process

There exist two alternative ways to study a sequence of event arrivals. First we can consider the sequence of arrival dates or equivalently the sequence of durations

Y1r…, Yn,… between consecutive events. Secondly, we can introduce the counting process [N(t), t varying], which counts the number of events observed between 0 and t. The counting process is a jump process, whose jumps of unitary size occur at each arrival date. It is equivalent to know the sequence of durations or the path of the counting process.

The Poisson process is obtained by imposing the following two conditions:

1. The counting process has independent increments, i. e. N(tn) – N(tn-1), N(tn-1) – N(tn-2),…, N(t1) – N(t0) are independent for any t0 < t1 < t2 < … < tn, and any n.

2. The rate of arrival of an event (a jump) is constant and two events cannot occur in a small time interval:

P[N(t + dt) – N(t) = 1] = Xdt + o(dt), (21.24)

P[N(t + dt) – N(t) = 0] = 1 – Xdt + o(dt), (21.25)

where the term o(dt) is a function of dt such that limdt^0 o(dt)/dt = 0.

Under these assumptions it is possible to deduce the distributions of the count­ing process and of the sequence of durations.

1. For a Poisson process, the durations Yi, i = 1,…, n are independent, identi­cally exponentially distributed with parameter X.

2. For a Poisson process, the increments N(t2) – N(t1), with t2 > t1, follow Poisson distributions, with parameters X(t2 – t1).

This result explains why basic duration models are based on exponential distri­butions, whereas basic models for count data are based on Poisson distributions, and how these specifications are related (see Cameron and Trivedi, Chapter 15, in this volume).

Similarly, a more complex dynamics can be obtained by relaxing the assump­tion that either the successive durations, or the increments of the counting pro­cess are independent. In the following subsections we introduce the temporal dependence duration sequences.