Lectures on the Poisson process

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