An introduction to the theory of Point Processes - Volume II: General theory and structure

Abstract

The long-awaited Volume II of the second edition of Daley and Vere-Jones' now classic book on point processes has appeared exactly 20 years after the first edition was published. It lives up to the very high standards of both the first edition [D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer, New York, 1988; MR0950166 (90e:60060)] and Volume I of the second [D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I, Second edition, Springer, New York, 2003; MR1950431 (2004c:60001)]. As stated by the authors, the second volume focuses on the structure and general theory of point processes, with the more elementary theory appearing in the first volume (Chapters 1--8). Volume II (Chapters 9--15) contains most of the material that appeared in Chapters 7--14 of the first edition, as well as a significant number of new topics. Chapter 9 presents a detailed treatment of the general theory of random measures and point processes, a topic touched on only briefly in Volume I. This is followed by Chapter 10, on special classes such as completely random measures and infinitely divisible point processes, which also includes new material on Markov properties of point processes on general metric spaces. Convergence concepts are introduced in Chapter 11. Stationary point processes and random measures are the subject of Chapter 12, with new material on long-range dependence and self-similarity. Chapter 13 is devoted to Palm theory and includes a new section on fractal dimensions. The martingale approach to marked point processes on {\Bbb R_+} is exploited in Chapter 14, on so-called ``evolutionary'' processes---i.e., processes that evolve in time. Finally, Chapter 15 focuses on particular properties of spatial and space-time point processes. As was the case with the first edition, each topic is carefully motivated and an extensive literature review is provided. Heuristic explanations accompany rigorous proofs, and numerous examples and exercises (without solutions) are given. While the more basic material in Volume I is more widely accessible, Volume II will be of particular use to the specialist with a good background in probability, measure theory, and topology. In his review of the first edition of this book [op. cit.; MR0950166 (90e:60060)], Alan Karr predicted that it was ``likely to become the reference on point process theory''. His prediction was correct; both volumes of the second edition belong on the shelf of any modern probability theorist.