Lectures on the Poisson Process

Front Cover
Cambridge University Press, Oct 26, 2017 - Mathematics - 310 pages
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Point Processes
9
The Mecke Equation and Factorial Measures
26
Mappings Markings and Thinnings
38
Characterisations of the Poisson Process
46
Poisson Processes on the Real Line
58
8
69
The Palm Distribution
82
Extra Heads and Balanced Allocations
92
Normal Approximation in the Boolean Model
227
Appendix A Some Measure Theory
239
Metric Spaces
250
Hausdorff Measures and Additive Functionals
252
Measures on the Real HalfLine
257
Absolutely Continuous Functions
259
Appendix B Some Probability Theory
261
Mean Ergodic Theorem
264

Stable Allocations
103
Random Measures and Cox Processes
127
Compound Poisson Processes
153
The Boolean Model and the Gilbert Graph
166
The Boolean Model with General Grains
179
Perturbation Analysis
197
Covariance Identities
211
The Central Limit Theorem and Steins Equation
266
Conditional Expectations
268
Gaussian Random Fields
269
Historical Notes
272
References
281
Index
289
Copyright

Other editions - View all

Common terms and phrases

About the author (2017)

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.