An Introduction to the Theory of Point Processes (eBook)

Preface to Volume II, Second Edition 7
Contents 9
Chapter Titles for Volume I 11
Principal Notation 12
Concordance of Statements from the First Edition 16
9 Basic Theory of Random Measures and Point Processes 18
9.1. Definitions and Examples 19
9.2. Finite-Dimensional Distributions and the Existence Theorem 42
9.3. Sample Path Properties: Atoms and Orderliness 55
9.4. Functionals: Definitions and Basic Properties 69
9.5. Moment Measures and Expansions of Functionals 82
10 Special Classes of Processes 93
10.1. Completely Random Measures 94
10.2. In.nitely Divisible Point Processes 104
10.3. Point Processes De.ned by Markov Chains 112
10.4. Markov Point Processes 135
11 Convergence Concepts and Limit Theorems 148
11.1. Modes of Convergence for Random Measures and Point Processes 149
11.2. Limit Theorems for Superpositions 163
11.3. Thinned Point Processes 172
11.4. Random Translations 183
12 Stationary Point Processes and Random Measures 193
12.1. Stationarity: Basic Concepts 194
12.2. Ergodic Theorems 211
12.3. Mixing Conditions 223
12.4. Stationary In.nitely Divisible Point Processes 233
12.5. Asymptotic Stationarity and Convergence to Equilibrium 239
12.6. Moment Stationarity and Higher- order Ergodic Theorems 253
12.7. Long-range Dependence 266
12.8. Scale-invariance and Self-similarity 272
13 Palm Theory 285
13.1. Campbell Measures and Palm Distributions 286
13.2. Palm Theory for Stationary Random Measures 301
13.3. Interval- and Point-stationarity 316
13.4. Marked Point Processes, Ergodic Theorems, and Convergence to Equilibrium 334
13.5. Cluster Iterates 351
13.6. Fractal Dimensions 357
14 Evolutionary Processes and Predictability 372
14.1. Compensators and Martingales 373
14.2. Campbell Measure and Predictability 393
14.3. Conditional Intensities 407
14.4. Filters and Likelihood Ratios 417
14.5. A Central Limit Theorem 429
14.6. Random Time Change 435
14.7. Poisson Embedding and Existence Theorems 443
14.8. Point Process Entropy and a Shannon – MacMillan Theorem 457
15 Spatial Point Processes 474
15.1. Descriptive Aspects: Distance Properties 475
15.2. Directional Properties and Isotropy 483
15.3. Stationary Line Processes in the Plane 488
15.4. Space–Time Processes 502
15.5. The Papangelou Intensity and Finite Point Patterns 523
15.6. Modi.ed Campbell Measures and Papangelou Kernels 535
15.7. The Papangelou Intensity Measure and Exvisibility 543
References with Index 554
Subject Index 574