We now start the study of a stochastic process which is absolutely vital for applications. Random number generation (Lu et al., 1996), queuing theory (Gross and Harris, 1998), quantum physics (Jona-Lasinio, 1985), electronics (Kingman, 1993), and biology (Blæsild and Granfeldt, 2002) are only some of the many areas of application for this process (Good, 1986).
The process is named in honor of Siméon-Denis Poisson (1781–1840), and, as we shall see, it is a continuous-time process (the index set ) with discrete state space .
We start with basic definitions.
As always, we work on a probability space endowed with a filtration . However, as we shall see, the Poisson process has a discrete state space. Therefore, as mentioned in the probability part of the book, the sigma algebra and the filtration are not essential, and we shall take formed from the pre-images of all subsets of . This will allow us to have every set measurable, ...