Lectures on the Poisson Process: (Series Number 7 Institute of Mathematical Statistics Textbooks)

The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.

Table of Contents:
Preface; List of symbols; 1. Poisson and other discrete distributions; 2. Point processes; 3. Poisson processes; 4. The Mecke equation and factorial measures; 5. Mappings, markings and thinnings; 6. Characterisations of the Poisson process; 7. Poisson processes on the real line; 8. Stationary point processes; 9. The Palm distribution; 10. Extra heads and balanced allocations; 11. Stable allocations; 12. Poisson integrals; 13. Random measures and Cox processes; 14. Permanental processes; 15. Compound Poisson processes; 16. The Boolean model and the Gilbert graph; 17. The Boolean model with general grains; 18. Fock space and chaos expansion; 19. Perturbation analysis; 20. Covariance identities; 21. Normal approximation; 22. Normal approximation in the Boolean model; Appendix A. Some measure theory; Appendix B. Some probability theory; Appendix C. Historical notes; References; Index.

About the Author :
Günter Last is Professor of Stochastics at the Karlsruhe Institute of Technology, Germany. He is a distinguished probabilist with particular expertise in stochastic geometry, point processes, and random measures. He coauthored a research monograph on marked point processes on the line as well as two textbooks on general mathematics. He has given many invited talks on his research worldwide. Mathew Penrose is Professor of Probability at the University of Bath. He is an internationally leading researcher in stochastic geometry and applied probability and is the author of the influential monograph Random Geometric Graphs (2003). He received the Friedrich Wilhelm Bessel Research Award from the Humboldt Foundation in 2008, and has held visiting positions as guest lecturer in New Delhi, Karlsruhe, San Diego, Birmingham, and Lille.

Review :
'An understanding of the remarkable properties of the Poisson process is essential for anyone interested in the mathematical theory of probability or in its many fields of application. This book is a lucid and thorough account, rigorous but not pedantic, and accessible to any reader familiar with modern mathematics at first degree level. Its publication is most welcome.' J. F. C. Kingman, University of Bristol 'I have always considered the Poisson process to be a cornerstone of applied probability. This excellent book demonstrates that it is a whole world in and of itself. The text is exciting and indispensable to anyone who works in this field.' Dietrich Stoyan, Technische Universität Bergakademie Freiberg , Germany 'Last and Penrose's Lectures on the Poisson Process constitutes a splendid addition to the monograph literature on point processes. While emphasizing the Poisson and related processes, their mathematical approach also covers the basic theory of random measures and various applications, especially to stochastic geometry. They assume a sound grounding in measure-theoretic probability, which is well summarized in two appendices (on measure and probability theory). Abundant exercises conclude each of the twenty-two 'lectures' which include examples illustrating their 'course' material. It is a first-class complement to John Kingman's essay on the Poisson process.' Daryl Daley, University of Melbourne 'Pick n points uniformly and independently in a cube of volume n in Euclidean space. The limit of these random configurations as n → ∞ is the Poisson process. This book, written by two of the foremost experts on point processes, gives a masterful overview of the Poisson process and some of its relatives. Classical tenets of the Theory, like thinning properties and Campbell's formula, are followed by modern developments, such as Liggett's extra heads theorem, Fock space, permanental processes and the Boolean model. Numerous exercises throughout the book challenge readers and bring them to the edge of current theory.' Yuval Peres, Microsoft Research and National Academy of Sciences 'The book under review fills an essential gap and is a very valuable addition to the point process literature. There is no doubt that this volume is a milestone and will very quickly become a standard reference in every field in which the Poisson process appears.' Christoph Thale, MathSciNet