Lectures on the Poisson ProcessThe 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. |
Contents
1 Poisson and Other Discrete Distributions | 1 |
2 Point Processes | 9 |
3 Poisson Processes | 19 |
4 The Mecke Equation and Factorial Measures | 26 |
5 Mappings Markings and Thinnings | 38 |
6 Characterisations of the Poisson Process | 46 |
7 Poisson Processes on the Real Line | 58 |
8 Stationary Point Processes | 69 |
15 Compound Poisson Processes | 153 |
16 The Boolean Model and the Gilbert Graph | 166 |
17 The Boolean Model with General Grains | 179 |
18 Fock Space and Chaos Expansion | 187 |
19 Perturbation Analysis | 197 |
20 Covariance Identities | 211 |
21 Normal Approximation | 219 |
22 Normal Approximation in the Boolean Model | 227 |
9 The Palm Distribution | 82 |
10 Extra Heads and Balanced Allocations | 92 |
11 Stable Allocations | 103 |
12 Poisson Integrals | 111 |
13 Random Measures and Cox Processes | 127 |
14 Permanental Processes | 136 |
Appendix A Some Measure Theory | 239 |
Appendix B Some Probability Theory | 261 |
Appendix C Historical Notes | 272 |
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Common terms and phrases
A(dx assertion assume assumption binomial Boolean model Borel space bounded compact compound Poisson process convex countable Cox process define Definition denote distribution Q factorial moment factorial moment measure finite measure follows formula Fubini's theorem generalised given Hence Hint implies inequality intensity measure kernel Lebesgue measure Lemma Let f Let m e locally finite measurable function measurable mapping measurable space Mecke equation metric space monotone obtain or-field or-finite P(n e point process Poisson distribution probability measure process with intensity Proof Let Proposition prove Q(dK random element random field random measure random variable resp result right-hand side satisfying sequence Show signed measure simple point process ſº ſſ stationary point stationary point process subset Suppose x e Rº yields