Poisson Processes: (3 Oxford Studies in Probability)

In the theory of random processes there are two that are fundamental, and occur over and over again, often in surprising ways. There is a real sense in which the deepest results are concerned with their interplay. One, the Bachelier Wiener model of Brownian motion, has been the subject of many books. The other, the Poisson process, seems at first sight humbler and less worthy of study in its own right. Nearly every book mentions it, but most hurry past to more general point processes or Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general context. This book attempts to redress the balance. It records Kingman's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful, and a few key results often produce surprising consequences.

Table of Contents:
Stochastic models for random sets of points; Poisson processes in general spaces; Sums over Poisson processes; Poisson processes on the line; Marked Poisson processes; Cox processes; Stochastic geometry; Complete random measures; The Poisson-Dirichlet distribution.

Review :
'provides an enjoyable and clearly written introduction to the structure and properties of Poisson processes ... Thanks to skillful steering away from, and around, technicalities, it is widely accessible. If you don't know the story, read this book - you will then know what you are missing. If you do know it, a browse through the book, particularly the later chapters, is still worthwhile for interesting perspectives on several areas.' P.J. Donnelly, Queen Mary and Westfield College, Short Book Reviews (Publication of the International Statistical Institute) 'It records the author's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions.' L'Enseignement Mathématique, 3-4, 1993 'Every mathematician with some knowledge of stochastic processes is aware of the interest and importance of the Poisson process. Therefore it is very useful to have now a book which is devoted to a systematic treatment of Poisson processes. The book ... fulfills the expectations one might have when a famous elder author writes a book on a classic topic. It gives the basic facts in a clear and lucid way. It is shown how the theory can be applied to interesting problems of astronomy, queueing and traffic, etc., and these examples are studied very thoroughly and deeply, giving even the specialist new insights ... an excellent basis for lectures or seminars .... a valuable gift for a young mathematician to stimulate his or her interest in stochastic processes and in applied probability in general.' Mathematical Reviews, Issue 94a 'The presentation everywhere is rigorous without being fuzzy about measure theoretical details; this would make the monograph suitable for many readrs, who are either not interested or not trained in measure theoretical subtleties ... 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.' S.D. Chatterji, Mathematics Abstracts, 773/93 `The Poisson process is surely the most beautiful object in probability theory, and John Kingman is its most gifted expositor. One might have been forgiven for thinking that there would be little new to say, but in fact this book is studded with new and fascinating insights. It is rare to find a book that simultaneously addresses the beginner and the expert. If there were a prize for the wisest probability book of the decade it would have to go to Bristol's Vice-Chancellor.' David Kendall, Cambridge, Journal of Royal Statistical Society, 1994