An introduction to the theory of point processes. Vol. II: General theory and structure.
2nd revised and extended ed.

*(English)*Zbl 1159.60003
Probability and Its Applications. New York, NY: Springer (ISBN 978-0-387-21337-8/hbk; 978-0-387-49835-5/ebook). xvii, 573 p. (2008).

In the presented research monograph the authors have made a major reshaping of their work in their first edition of 1988 [An introduction to the theory of point processes. Springer (1988; Zbl 0657.60069)] and now present An Introduction to the Theory of Point Processes in two volumes with subtitles Volume I: Elementary Theory and Methods and Volume II: General Theory and Structure.

Volume I [New York, NY: Springer (2003; Zbl 1026.60061)] consists of 8 chapters, including standard material: Poisson process, stationary point processes on the line, renewal processes, finite point processes, Cox, cluster and marked point processes, conditional intensities and likelihood and second order properties of stationary point processes. In the three appendices basic concepts of topology and measure theory, measures of metric spaces and conditional expectations, stopping times and martingales are reviewed, where the latter underlies the discussion of predictability and conditional intensity of Volume II, Chapter 14.

In the second volume they set out a general unified framework for the theory of point processes by starting from their interpretation as random measures. The material represents a reorganized version of those parts of Chapter 6–14 of the first edition, not already covered in Volume I, continued with the more theoretical topics of the first edition, including limit theorems, ergodic theory, Palm theory and evolutionary behavior via martingales and conditional intensity and the structure of spatial point processes.

Readers of the second volume should be more familiar with aspects of measure theory and topology than those of the first volume. The second volume starts with Chapter 9: Basic theory of random measures and point processes. Then (Chapter 10) special classes of processes are discussed like infinite divisible point processes, point processes defined by Markov chains and marked point processes. Chapter 11 deals with convergence concepts and limit theorems, including superpositions, thinning and random transitions. Stationary point processes and random measures is the topic of Chapter 12: ergodic theorems, infinite divisible point processes, asymptotic stationarity and convergence to equilibrium, long range dependence, self similarity. The Palm theory is treated extensively in Chapter 13. In Chapter 14 a general introduction to the notions of compensators and point process martingales and their links to the Doob-Meyer theorems are given. Further extensions to random measures and marked point processes are given. Then conditional intensities are treated as a Radon-Nikodym derivative of Campbell measures are re-introduced. Finally various applications are given (likelihood and time change theorems, martingale-type central limit theorem, entropy). Spatial point processes (in \(\mathbb R^2\) and \(\mathbb R^3\)) are dealt in the last Chapter 15: In the first four sections the authors review mainly descriptive poverties, distinguishing between distance and directional properties of spatial point patterns, starting from finite models, moving to the moment properties of line processes, and then revisiting space-time models. In the final section is given an introduction to modeling centered around the concept of Papangelou intensity. Some background and motivation from statistical and physical settings is provided and then an introduction to the more mathematical theory is given (modified Campbell measures, Papangelou kernels, Papangelou intensity, and exvisibility).

The comprehensive unified exposition of known and substantial new material given in this second volume will make this research monograph to a standard text book on advanced subjects of point processes. The many examples illustrate the mathematical results and the various applicability of point processes in various fields.

Volume I [New York, NY: Springer (2003; Zbl 1026.60061)] consists of 8 chapters, including standard material: Poisson process, stationary point processes on the line, renewal processes, finite point processes, Cox, cluster and marked point processes, conditional intensities and likelihood and second order properties of stationary point processes. In the three appendices basic concepts of topology and measure theory, measures of metric spaces and conditional expectations, stopping times and martingales are reviewed, where the latter underlies the discussion of predictability and conditional intensity of Volume II, Chapter 14.

In the second volume they set out a general unified framework for the theory of point processes by starting from their interpretation as random measures. The material represents a reorganized version of those parts of Chapter 6–14 of the first edition, not already covered in Volume I, continued with the more theoretical topics of the first edition, including limit theorems, ergodic theory, Palm theory and evolutionary behavior via martingales and conditional intensity and the structure of spatial point processes.

Readers of the second volume should be more familiar with aspects of measure theory and topology than those of the first volume. The second volume starts with Chapter 9: Basic theory of random measures and point processes. Then (Chapter 10) special classes of processes are discussed like infinite divisible point processes, point processes defined by Markov chains and marked point processes. Chapter 11 deals with convergence concepts and limit theorems, including superpositions, thinning and random transitions. Stationary point processes and random measures is the topic of Chapter 12: ergodic theorems, infinite divisible point processes, asymptotic stationarity and convergence to equilibrium, long range dependence, self similarity. The Palm theory is treated extensively in Chapter 13. In Chapter 14 a general introduction to the notions of compensators and point process martingales and their links to the Doob-Meyer theorems are given. Further extensions to random measures and marked point processes are given. Then conditional intensities are treated as a Radon-Nikodym derivative of Campbell measures are re-introduced. Finally various applications are given (likelihood and time change theorems, martingale-type central limit theorem, entropy). Spatial point processes (in \(\mathbb R^2\) and \(\mathbb R^3\)) are dealt in the last Chapter 15: In the first four sections the authors review mainly descriptive poverties, distinguishing between distance and directional properties of spatial point patterns, starting from finite models, moving to the moment properties of line processes, and then revisiting space-time models. In the final section is given an introduction to modeling centered around the concept of Papangelou intensity. Some background and motivation from statistical and physical settings is provided and then an introduction to the more mathematical theory is given (modified Campbell measures, Papangelou kernels, Papangelou intensity, and exvisibility).

The comprehensive unified exposition of known and substantial new material given in this second volume will make this research monograph to a standard text book on advanced subjects of point processes. The many examples illustrate the mathematical results and the various applicability of point processes in various fields.

Reviewer: Andreas Brandt (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60G57 | Random measures |