This book provides an introduction to the theory and applications of point processes, both in time and in space. Presenting the two components of point process calculus, the martingale calculus and the Palm calculus, it aims to develop the computational skills needed for the study of stochastic models involving point processes, providing enough of the general theory for the reader to reach a technical level sufficient for most applications.
Classical and not-so-classical models are examined in detail, including Poisson–Cox, renewal, cluster and branching (Kerstan–Hawkes) point processes.The applications covered in this text (queueing, information theory, stochastic geometry and signal analysis) have been chosen not only for their intrinsic interest but also because they illustrate the theory.
Written in a rigorous but not overly abstract style, the book will be accessible to earnest beginners with a basic training in probability but will also interest upper graduate students and experienced researchers.
Author(s): Pierre Brémaud
Series: Probability Theory and Stochastic Modelling 98
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 556
Tags: Point Process, Poisson Process, Renewal Process, Branching Process, Palm Probability, Spectral Measure, Queueing Theory
Preface
Contents
Chapter 1 Generalities
1.1 Point Processes as Random Measures
1.2 Campbell’s Formula and Moment Measures
1.3 The Distribution of a Point Process
1.4 Convergence in Distribution and Variation
1.5 Cluster Point Processes
1.6 The Stieltjes–Lebesgue Calculus
1.7 Exercises
Chapter 2 Poisson Processes on the Line
2.1 Counting Process and Interval Sequence
2.2 The Smoothing Formula
2.3 Poisson Martingales and Stochastic Integrals
2.4 Watanabe’s Characterization
2.5 HMCs and Stochastic Differential Equations
2.6 HMCs and Time-Scaled HPPs
2.7 Exercises
Chapter 3 Spatial Poisson Processes
3.1 Sampling a Poisson Process
3.2 The Covariance and Exponential Formulas
3.3 Marked Spatial Poisson Processes
3.4 Operations on Poisson Processes
3.5 Change of Probability
3.6 Exact Sampling of Cluster Point Processes
3.7 The Boolean Model
3.8 Exercises
Chapter 4 Renewal and Regenerative Processes
4.1 Renewal Point processes
4.2 The Renewal Theorem
4.3 Blackwell’s Theorem and its Refinements
4.4 Regenerative Processes
4.5 Multivariate Renewal Equations
4.6 Semi-Markov Processes
4.7 Exercises
Chapter 5 Point Processes with a Stochastic Intensity
5.1 The Smoothing Formulas
5.2 Regenerative Form of the Intensity Kernel
5.3 Martingales as Stochastic Integrals
5.4 Time Scaling
5.5 Continuous Change of Probability
5.6 Extension of the Theory of Stochastic Intensity
5.7 Grigelionis’ Representation
5.8 Origin and Motivation of the Martingale Approach
5.9 Exercises
Chapter 6 Exvisible Intensity of Finite Point Processes
6.1 The Janossy Density
6.2 The Spatial Smoothing formula
6.3 Exvisibility and Predictability
6.4 Finite Markov Point Processes
6.5 Spatial Birth-and-Death Point Processes
6.6 An Alternative Model
6.7 Exercises
Chapter 7 Palm Probability on the Line
7.1 Stationary Point Processes
7.2 A First Look at Palm Probability
7.3 Palm Theory on the Line: Basic Formulas
7.4 From Palm Probability to Stationary Probability
7.5 Local Interpretation of Palm Probability
7.6 The Cross-ergodic Theorem
7.7 Palm Probability and Stochastic Intensity
7.8 Exercises
Chapter 8 Palm Probability in Space
8.1 The Voronoi Cell and the Inversion Formula
8.2 The Local Interpretation
8.3 Ergodicity
8.4 The Mecke Measure
8.5 The Reduced Mecke Measure
8.6 Exercises
Chapter 9 The Power Spectral Measure
9.1 The Covariance Measure
9.2 The Bartlett Spectral Measure
9.3 Long-range Dependence
9.4 Transformations of the Spectral Measure
9.5 Exercises
Chapter 10 The Information Content of Point Processes
10.1 Filtering
10.2 Separation of Detection and Filtering
10.3 Hiding Information in a Point Process
10.4 Noisy Points
10.5 Random Sampling
10.6 Exercises
Chapter 11 Point Processes in Queueing
11.1 A Review of Markovian Queueing Theory
11.2 Poisson Systems
11.3 The G/G/1/∞ Queue
11.4 PASTA, Little, etc.
11.5 Selected Applications
11.6 Exercises
Chapter 12 Hawkes Point Processes
12.1 As a Branching Point Process
12.2 Rates of Extinction and of Installation
12.3 The Bartlett Spectrum of the Hawkes Process
12.4 Exact Sampling of Hawkes Processes
12.5 Branching Point Processes Without Ancestor
12.6 Kerstan Point Processes
12.7 Exercises
Appendix
A.1 Measurability and Measure
A.2 Stochastic Processes
A.3 Martingales
A.4 Internal History of a Marked Point Process
A.5 Local vs. Global Absolute Continuity
Bibliography
Index
