Point Processes

Point Processesis a book on the mathematics ofpoint processes,randomly located sets of points on thereal lineor in other geometric spaces. It was written byDavid CoxandValerie Isham,and published in 1980 byChapman & Hallin their Monographs on Applied Probability and Statistics book series. The Basic Library List Committee of theMathematical Association of Americahas suggested its inclusion in undergraduate mathematics libraries.

Topics

AlthoughPoint Processescovers some of the general theory of point processes, that is not its main focus, and it avoids any discussion ofstatistical inferenceinvolving these processes. Instead, its aim is to present the properties and descriptions of several specific processes arising in applications of this theory,which had not been previously collected in texts in this area.

Three of its six chapters concern more general material, while the final three are more specific. The first chapter includes introductory material on standard processes:Poisson point processes,renewal processes,self-exciting processes,anddoubly stochastic processes.The second chapter provides some general theory includingstationarity,orderliness (meaning that the probability of multiple arrivals in short intervals is sublinear in the interval length),Palm distributions,Fourier analysis,andprobability-generating functions.Chapter four (the third of the more general chapters) concernspoint process operations,methods of modifying or combining point processes to generate other processes.

Chapter three, the first of the three chapters on more specific models, is titled "Special models".The special models that it covers include non-stationary Poisson processes,compound Poisson processes,and theMoran process,along with additional treatment of doubly stochastic processes and renewal processes. Until this point, the book focuses on point processes on the real line (possibly also with a time dimension), but the two final chapters concernmultivariateprocesses and on point processes for higher dimensional spaces, including spatio-temporal processes andGibbs point processes.

Audience and reception

The book is primarily a reference for researchers.It could also be used to provide additional examples for a course onstochastic processes,or as the basis for an advanced seminar. Although it uses relatively little advanced mathematics, readers are expected to understand advanced calculus and have some familiarity withprobability theoryandMarkov chains.

Writing some ten years after its original publication, reviewer Fergus Daly ofThe Open Universitywrites that his copy has been well used, and that it "still is a very good book: lucid, relevant and still not matched in its approach by any other text".