Poisson Processes [Hardcover]

Two fundamental theories are commonly debated in the study of random processes: the Bachelier Wiener model of Brownian motion, which has been the subject of many books, and the Poisson process. While nearly every book mentions the Poisson process, most hurry past to more general point processes or to Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general contexts. This book attempts to redress the balance. It records the author's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful and a few key results often produce surprising consequences.

1. Stochastic Models for Random Sets of Points
2. Poisson Processes in General Spaces
3. Sums Over Poisson Processes
4. Poisson Processes on the Line
5. Marked Poisson Processes
6. Cox Processes
7. Stochastic Geometry
8. Complete Random Measures
9. The Poisson-Dirichlet Distribution.

Fulfills the expectations one might have when a famous elder author writes a book on a classic topic. It gives the basic facts in a clear and lucid way. It is shown how the theory can be applied to interesting problems of astronomy, queuing and traffic, etc., and these examples are studied very thoroughly and deeply, giving even the specialist new insights . . . . an excellent basis for lectures or seminars . . . a valuable gift for a young mathematician to stimulate his or her interest in stochastic processes and in applied probability in general. --Dietrich Stoyan, MathematicalReviews

Whereas treatises on Brownian motion abound, there is a dearth of parallel treatments of the Poisson process. In point of fact, this book is the very first. We read it from cover to cover as soon as we receiló“