Poisson processes. (English) Zbl 0771.60001

Kingman, J. F. C.

Oxford Studies in Probability. 3. Oxford: Clarendon Press. viii, 104 p. (1993).

This monograph is an introductory account of the modern theory of Poisson processes and its many applications. The titles of the different chapters give a rough description of its contents: 1. Stochastic models for random sets of points. 2. Poisson processes in general spaces. 3. Sums over Poisson processes. 4. Poisson processes on the line. 5. Marked Poisson processes. 6. Cox processes. 7. Stochastic geometry. 8. Completely random measures. 9. The Poisson-Dirichlet distribution.
After an introductory chapter containing, amongst other things, various fundamental properties of the usual (\(\mathbb{N}\)-valued) Poisson random variables the author introduces in the second chapter the general definition of a Poisson process on a measurable state space \(S\) (supposed to satisfy a fairly weak hypothesis, verified if \(S=\mathbb{R}^ d\) or any other separable metric space): the process is defined by one “random variable” \(\Pi\), defined on some probability space \(\Omega\), whose values \(\Pi(\omega)\), \(\omega\in\Omega\), are denumerable subsets of \(S\) such that for each measurable subset \(A\subset S\), the quantity \(\#\{\Pi(\omega)\cap A\}=N(A)(\omega)\) \((\#\) representing “cardinality”) is a usual \(\overline\mathbb{N}\)-valued random variable; two hypotheses are then imposed on the stochastic process \(\{N(A)\}_{A\subset S}\) (\(A\) measurable) obtained from the “count functions” \(N(A)\): (i) \(N(A_ 1),\dots,N(A_ n)\) are independent if \(A_ 1,\dots,A_ n\) are disjoint; (ii) \(N(A)\) has the Poisson distribution with parameter \(\mu(A)\in[0,\infty]\).
The basic properties of such Poisson processes are then given in Chapters 2, 3, 4, notably, a clear discussion of Campbell’s theorem (concerning the random variable \(\sum_{X\in\Pi}f(X)\), \(f\) being a real-valued function on the state space \(S)\) and a complete proof of a version of an elegant theorem of Rényi which asserts that if \(\mathbb{P}\text{rob}(N(A)=0)\) is of the form \(\exp(-\mu(A))\) (for a large class of sets \(A\) and \(\mu\) nonatomic), then the \(N(A)\)’s are automatically independent. The first four chapters form about half the book; in the last four chapters processes are treated which are either more general than Poisson processes or else specialisations obtained for various applications as for instance in stochastic geometry, ecology, traffic problems, or queueing theory (already introduced in Chapter 4).
The presentation everywhere is rigorous without being fuzzy about measure theoretical details; this would make the monograph suitable for many readers, who are either not interested or not trained in measure theoretical subtelities. In the last four chapters, the author introduces a variety of topics without going into all the proofs or details; however, he does provide references which would permit readers to pursue matters further. All in all, the monograph is a useful addition to the literature both for various beginners as well as for lecturers in the theory of stochastic processes who would find in it a rich array of topics presented clearly.

Reviewer: S.D.Chatterji (Lausanne)

MSC:

60-02

Research exposition (monographs, survey articles) pertaining to probability theory

Keywords:

stochastic geometry; marked Poisson processes; modern theory of Poisson processes; Cox processes; random measures; Poisson-Dirichlet distribution; Campbell’s theorem