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Random point processes in time and space. 2nd ed. (English) Zbl 0744.60050
Springer Texts in Electrical Engineering. New York etc.: Springer-Verlag. xi, 481 p. (1991).
This book is a revision of the book “Random point processes” (1975; Zbl 0385.60052). Much more emphasis is given to point processes in two dimensions, which reflect the tremendous increase that has taken place in the use of point-process models for the description of data in a wide variety of scientific and engineering disciplines. Several chapters of the first edition have been modified, a new chapter on translated Poisson processes has been added and the Chapter 7 on marked point-processes has been eliminated to make room for new material.
In Chapter 2 the Poisson process in time is defined, its properties are investigated, procedures for its simulation are outlined and the important practical problem of estimating parameters is discussed.
Translated point processes are developed in Chapter 3. These point processes result by translating each point of a Poisson process by some random amount to another location. The model includes insertions and deletions of points. Special attention is given to the estimation of parameters concerning the original Poisson process given data determined by points of the translated Poisson process.
Compound Poisson processes are studied in Chapter 4. These processes are obtained from a given Poisson process by associating an auxiliary random mark with each point. It is shown that a compound Poisson process can be represented as a superposition of independent Poisson counting-processes. This result is used to specify procedures for estimating unknown parameters that influence an observed compound Poisson process.
In Chapter 5 filtered Poisson processes and Poisson-driven Markov processes are studied. These processes are the response of linear and nonlinear dynamic systems to compound Poisson processes. In the linear case the response is a superposition of the individual responses to each marked point. Statistics for such processes are developed and applications to shot noise and photoelectron noise are given. The main mathematical tool in the nonlinear case is the theory of stochastic differential and integral equations, which is also developed in some detail.
Self-exciting point processes are studied in Chapter 6. New results are presented for estimating hazard functions, which has been motivated by recently developed models of neural discharge.
Finally an investigation of doubly stochastic processes is given in Chapter 7. Such processes are reasonable models for photoelectric emissions. Algorithms for the estimation of random signals that influence the underlying intensity with which the observed points occur are developed.
At the end of each chapter numerous very interesting and illustrative problems are posed and many references are given. This book is a must to all, who are interested in the application of point processes.
Reviewer: P.Weiß (Linz)

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory