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Lectures on the Poisson process. (English) Zbl 1392.60004
Institute of Mathematical Statistics Textbooks 7. Cambridge: Cambridge University Press (ISBN 978-1-107-45843-7/pbk; 978-1-107-08801-6/hbk; 978-1-316-10447-7/ebook). xx, 293 p. (2018).
This is a modern textbook on general Poisson processes, with applications mostly to stochastic geometry. The book can serve both for graduate courses or seminars on the Poisson process and as an introduction to the theory of point processes. The book contains 22 chapters and 3 appendices. Principally, each chapter can be a base for one or two lectures. Chapters 1–7 contain fundamentals of Poisson processes, including the Mecke equation and factorial measures, mappings and restrictions, the marking theorem, Rényi’s theorem. Chapters 8–11, 16–17 and 22 are devoted to stochastic geometry. Topics as stationary point processes including ergodicity, the Palm distribution, balanced and stable allocations, the Boolean model and the Gilbert graph are considered. In other chapters, more advanced results on the stochastic analysis of Poisson process are presented, including the Fock space and chaos expansion, perturbation analysis and normal approximation. The book is self-contained and the reader only needs to be acquainted with measure-theoretic probability theory. The book is written in a lively and interesting style for the reader, while it retains the full formality of the presentation. It can be recommended for graduate and post-graduate students, teachers and specialists in stochastic processes and its applications, as well as for all those who wish to get acquainted with such a modern and applicable object as the Poisson process.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G10 Stationary stochastic processes
60D05 Geometric probability and stochastic geometry
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