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**An introduction to the theory of point processes.**
*(English)*
Zbl 0657.60069

Springer Series in Statistics. New York etc.: Springer-Verlag. xxi, 702 p. DM 145.00 (1988).

This book provides a general introduction to the theory of point processes at an intermediate level between more abstract theoretical treatments such as those in the monographs of J. Kerstan, K. Matthes and J. Mecke [Infinitely divisible point processes (1978; Zbl 0383.60001)], or O. Kallenberg [Random measures (1983; Zbl 0544.60053)] and more informal treatments dealing mainly with applications such as those in the monographs of D. R. Cox and P. A. W. Lewis [The statistical analysis of series of events (1966; Zbl 0148.140)], D. L. Snyder [Random point processes (1975; Zbl 0385.60052)] or D. R. Cox and V. Isham [Point processes (1980; Zbl 0441.60053)].

Chapter 1 is entitled “Early history” - quite uncommon for a mathematics text. Here the mainstreams of the historical development are described which led to the modern mathematical theory, beginning with its roots in such diverse subjects as life tables and the theory of self- renewing aggregates, counting problems, particle physics and population processes, and communication engineering.

In Chapters 2-4 basic special cases of point processes are discussed, namely, “Poisson processes”, “Simple results for stationary point processes on the line” and “Renewal processes”. Throughout these chapters, with the exception of Section 2.4 on the general Poisson process, the state space is the real line. The focus is on an exposition of the more elementary probabilistic properties of these fundamental examples which are an illustration of and a motivation for more general results, with more sophisticated measure-theoretic problems being postponed to Chapters 6 and 7.

Chapter 5 on “Finite point processes” prepares the reader for the study of point process theory in more general state spaces than the line. Accordingly, Chapters 6-12 contain a thorough development of point process theory in the context of complete separable metric spaces, which constitutes the heart of the book. The necessary prerequisites from topology and measure theory are collected in Appendix 1, mostly without proofs, whereas in Appendix 2 the main results needed from the theory of measures on complete separable metric spaces and their convergence are given, including complete proofs.

Chapters 6 and 7 are an “Introduction to the general theory of random measures” and an “Introduction to the general theory of point processes”, respectively, providing the basic notions and facts such as existence theorems, sample-path properties, random integrals, characteristic and probability generating functionals, and moment and factorial moment measures.

Chapter 8 on “Cluster processes, infinitely divisible processes, and doubly stochastic processes” is devoted to the study of these important classes of point processes, mainly by the probability generating functional techniques introduced in Chapter 7.

Further specific aspects of the theory developed in Chapters 9-12 are “Convergence concepts and limit theorems”, “Stationary point processes and random measures”, “Spectral theory”, and “Palm theory”.

Chapter 13 on “Conditional intensities and likelihoods” is an introduction to the martingale approach to statistical inference for point processes on the positive half-line. Here the presentation is supported by Appendix 3 containing the required tools from martingale theory and the general theory of stochastic processes. More detailed information about this important branch of point process theory developed during the last two decades is provided, for example, by the monographs of P. Brémaud [Point processes and queues. Martingale dynamics (1981; Zbl 0478.60004)], M. Jacobsen [Statistical analysis of counting processes (1982; Zbl 0518.60065)] or A. F. Karr [Point processes and their statistical inference (1986; Zbl 0601.62120)]; for the application of point processes in extreme value theory see S. I. Resnick [Extreme values, regular variation and point processes (1987; Zbl 0633.60001)].

The final Chapter 14 entitled “Exterior conditioning” briefly discusses the kind of dual to the Palm theory which is concerned with point processes under conditioning on the behaviour outside of bounded sets and which is connected with questions from statistical mechanics.

The present book is a most valuable survey of the mathematical theory of point processes and will be appreciated by everyone who is interested in the subject. As one of its main features all the presented material is put into a historical perspective, not only by Chapter 1 but throughout the whole text by many detailed discussions of its relationships with the existing literature. This will be extremely useful especially for the beginner on his or her way into this broad field.

The exercises range from simple examples to advanced illustrations and extensions of the theory, often supplied with hints or references to the literature. Many examples in the text link the mathematical theory with its applications. Therefore, the book is also a suitable source for workers in applied fields who have to use point process methodology and wish to learn about its mathematical background.

Chapter 1 is entitled “Early history” - quite uncommon for a mathematics text. Here the mainstreams of the historical development are described which led to the modern mathematical theory, beginning with its roots in such diverse subjects as life tables and the theory of self- renewing aggregates, counting problems, particle physics and population processes, and communication engineering.

In Chapters 2-4 basic special cases of point processes are discussed, namely, “Poisson processes”, “Simple results for stationary point processes on the line” and “Renewal processes”. Throughout these chapters, with the exception of Section 2.4 on the general Poisson process, the state space is the real line. The focus is on an exposition of the more elementary probabilistic properties of these fundamental examples which are an illustration of and a motivation for more general results, with more sophisticated measure-theoretic problems being postponed to Chapters 6 and 7.

Chapter 5 on “Finite point processes” prepares the reader for the study of point process theory in more general state spaces than the line. Accordingly, Chapters 6-12 contain a thorough development of point process theory in the context of complete separable metric spaces, which constitutes the heart of the book. The necessary prerequisites from topology and measure theory are collected in Appendix 1, mostly without proofs, whereas in Appendix 2 the main results needed from the theory of measures on complete separable metric spaces and their convergence are given, including complete proofs.

Chapters 6 and 7 are an “Introduction to the general theory of random measures” and an “Introduction to the general theory of point processes”, respectively, providing the basic notions and facts such as existence theorems, sample-path properties, random integrals, characteristic and probability generating functionals, and moment and factorial moment measures.

Chapter 8 on “Cluster processes, infinitely divisible processes, and doubly stochastic processes” is devoted to the study of these important classes of point processes, mainly by the probability generating functional techniques introduced in Chapter 7.

Further specific aspects of the theory developed in Chapters 9-12 are “Convergence concepts and limit theorems”, “Stationary point processes and random measures”, “Spectral theory”, and “Palm theory”.

Chapter 13 on “Conditional intensities and likelihoods” is an introduction to the martingale approach to statistical inference for point processes on the positive half-line. Here the presentation is supported by Appendix 3 containing the required tools from martingale theory and the general theory of stochastic processes. More detailed information about this important branch of point process theory developed during the last two decades is provided, for example, by the monographs of P. Brémaud [Point processes and queues. Martingale dynamics (1981; Zbl 0478.60004)], M. Jacobsen [Statistical analysis of counting processes (1982; Zbl 0518.60065)] or A. F. Karr [Point processes and their statistical inference (1986; Zbl 0601.62120)]; for the application of point processes in extreme value theory see S. I. Resnick [Extreme values, regular variation and point processes (1987; Zbl 0633.60001)].

The final Chapter 14 entitled “Exterior conditioning” briefly discusses the kind of dual to the Palm theory which is concerned with point processes under conditioning on the behaviour outside of bounded sets and which is connected with questions from statistical mechanics.

The present book is a most valuable survey of the mathematical theory of point processes and will be appreciated by everyone who is interested in the subject. As one of its main features all the presented material is put into a historical perspective, not only by Chapter 1 but throughout the whole text by many detailed discussions of its relationships with the existing literature. This will be extremely useful especially for the beginner on his or her way into this broad field.

The exercises range from simple examples to advanced illustrations and extensions of the theory, often supplied with hints or references to the literature. Many examples in the text link the mathematical theory with its applications. Therefore, the book is also a suitable source for workers in applied fields who have to use point process methodology and wish to learn about its mathematical background.

Reviewer: E.Häusler

### MSC:

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

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

60G57 | Random measures |