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

Contents......Page 10 Preface......Page 16 List of Symbols......Page 20 1.1 The Poisson Distribution......Page 22 1.2 Relationships Between Poisson and Binomial Distributions......Page 24 1.3 The Poisson Limit Theorem......Page 25 1.4 The Negative Binomial Distribution......Page 26 1.5 Exercises......Page 28 2.1 Fundamentals......Page 30 2.2 Campbell’s Formula......Page 33 2.3 Distribution of a Point Process......Page 35 2.4 Point Processes on Metric Spaces......Page 37 2.5 Exercises......Page 39 3.1 Definition of the Poisson Process......Page 40 3.2 Existence of Poisson Processes......Page 41 3.3 Laplace Functional of the Poisson Process......Page 44 3.4 Exercises......Page 45 4.1 The Mecke Equation......Page 47 4.2 Factorial Measures and the Multivariate Mecke Equation......Page 49 4.3 Janossy Measures......Page 53 4.4 Factorial Moment Measures......Page 55 4.5 Exercises......Page 57 5.1 Mappings and Restrictions......Page 59 5.2 The Marking Theorem......Page 60 5.3 Thinnings......Page 63 5.4 Exercises......Page 65 6.1 Borel Spaces......Page 67 6.2 Simple Point Processes......Page 70 6.3 Renyi’s Theorem......Page 71 6.4 Completely Orthogonal Point Processes......Page 73 6.5 Turning Distributional into Almost Sure Identities......Page 75 6.6 Exercises......Page 77 7.1 The Interval Theorem......Page 79 7.2 Marked Poisson Processes......Page 82 7.3 Record Processes......Page 84 7.4 Polar Representation of Homogeneous Poisson Processes......Page 86 7.5 Exercises......Page 87 8.1 Stationarity......Page 90 8.2 The Pair Correlation Function......Page 92 8.3 Local Properties......Page 95 8.4 Ergodicity......Page 96 8.5 A Spatial Ergodic Theorem......Page 98 8.6 Exercises......Page 101 9.1 Definition and Basic Properties......Page 103 9.2 The Mecke–Slivnyak Theorem......Page 105 9.3 Local Interpretation of Palm Distributions......Page 106 9.4 Voronoi Tessellations and the Inversion Formula......Page 108 9.5 Exercises......Page 110 10.1 The Extra Head Problem......Page 113 10.2 The Point-Optimal Gale–Shapley Algorithm......Page 116 10.3 Existence of Balanced Allocations......Page 118 10.4 Allocations with Large Appetite......Page 120 10.6 Exercises......Page 122 11.1 Stability......Page 124 11.3 Optimality of the Gale–Shapley Algorithms......Page 125 11.4 Uniqueness of Stable Allocations......Page 128 11.5 Moment Properties......Page 129 11.6 Exercises......Page 130 12.1 The Wiener–Ito Integral......Page 132 12.2 Higher Order Wiener–Ito Integrals......Page 135 12.3 Poisson U-Statistics......Page 139 12.4 Poisson Hyperplane Processes......Page 143 12.5 Exercises......Page 145 13.1 Random Measures......Page 148 13.2 Cox Processes......Page 150 13.3 The Mecke Equation for Cox Processes......Page 152 13.4 Cox Processes on Metric Spaces......Page 153 13.5 Exercises......Page 154 14.1 Definition and Uniqueness......Page 157 14.2 The Stationary Case......Page 159 14.3 Moments of Gaussian Random Variables......Page 160 14.4 Construction of Permanental Processes......Page 162 14.5 Janossy Measures of Permanental Cox Processes......Page 166 14.6 One-Dimensional Marginals of Permanental Cox Processes......Page 168 14.7 Exercises......Page 172 15.1 Definition and Basic Properties......Page 174 15.2 Moments of Symmetric Compound Poisson Processes......Page 178 15.3 Poisson Representation of Completely Random Measures......Page 179 15.4 Compound Poisson Integrals......Page 182 15.5 Exercises......Page 184 16.1 Capacity Functional......Page 187 16.2 Volume Fraction and Covering Property......Page 189 16.3 Contact Distribution Functions......Page 191 16.4 The Gilbert Graph......Page 192 16.5 The Point Process of Isolated Nodes......Page 197 16.6 Exercises......Page 198 17.1 Capacity Functional......Page 200 17.2 Spherical Contact Distribution Function and Covariance......Page 203 17.3 Identifiability of Intensity and Grain Distribution......Page 204 17.4 Exercises......Page 206 18.1 Difference Operators......Page 208 18.2 Fock Space Representation......Page 210 18.3 The Poincare ́ Inequality......Page 214 18.4 Chaos Expansion......Page 215 18.5 Exercises......Page 216 19.1 A Perturbation Formula......Page 218 19.2 Power Series Representation......Page 221 19.3 Additive Functions of the Boolean Model......Page 224 19.4 Surface Density of the Boolean Model......Page 227 19.5 Mean Euler Characteristic of a Planar Boolean Model......Page 228 19.6 Exercises......Page 229 20.1 Mehler’s Formula......Page 232 20.2 Two Covariance Identities......Page 235 20.4 Exercises......Page 238 21.1 Stein’s Method......Page 240 21.2 Normal Approximation via Difference Operators......Page 242 21.3 Normal Approximation of Linear Functionals......Page 246 21.4 Exercises......Page 247 22.1 Normal Approximation of the Volume......Page 248 22.2 Normal Approximation of Additive Functionals......Page 251 22.3 Central Limit Theorems......Page 256 22.4 Exercises......Page 258 A.1 General Measure Theory......Page 260 A.2 Metric Spaces......Page 271 A.3 Hausdorff Measures and Additive Functionals......Page 273 A.4 Measures on the Real Half-Line......Page 278 A.5 Absolutely Continuous Functions......Page 280 B.1 Fundamentals......Page 282 B.2 Mean Ergodic Theorem......Page 285 B.3 The Central Limit Theorem and Stein’s Equation......Page 287 B.4 Conditional Expectations......Page 289 B.5 Gaussian Random Fields......Page 290 Appendix C Historical Notes......Page 293 References......Page 302 Index......Page 310 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 This self-contained introduction to the Poisson process covers basic theory and certain advanced topics in the setting of a general abstract measure space. The text includes applications and numerous exercises, and is ideal for graduate courses or self-study by mathematicians, physicists, and engineers.