Lectures on Point-processes, Random measures, and stochastic geometry

Point-Processes

I. Random Measures and Point Processes

1. Foundations

This lecture offers an in-depth exploration of random measures and point processes, which are foundational concepts in the study of stochastic processes. A point process can be viewed as a random object that takes values in locally finite configurations of points or, equivalently, counting measures. We extend this concept to the more general notion of a random measure, which is a random object that takes measures as possible realizations.

We begin by establishing the necessary framework, including key structures such as mean measures, Laplace transforms, and void probabilities. This foundational section sets the stage for understanding the fundamental properties and behaviors of random measures and point processes.

Next, we delve into the characterization of distributions. This part of the lecture explores powers and moment measures, providing a detailed examination of the Laplace transform and its role in characterizing distributions. We also address the concept of independence in the context of random measures, which is crucial for understanding the interactions between different random measures.

The lecture then introduces stochastic integrals, equipping readers with the tools to integrate with respect to random measures. We also discuss the vague topology on the space of measures, which is essential for understanding convergence properties.

Finally, we provide a detailed examination of point processes. This includes an exploration of simple point processes, methods for enumerating points, and the generating function. We also cover factorial powers and factorial moment measures, which are crucial for understanding higher-order properties of point processes.

This comprehensive approach ensures a thorough understanding of both random measures and point processes, equipping readers with the foundational knowledge needed to navigate this complex field.

Contents:
1 Framework
2 Mean Measure, Laplace Transform and Void Probability
    2.1 Campbell’s Averaging Formula
3 Distribution Characterization
4 Distribution Characterization
    4.1 Powers and Moment Measures
    4.2 Laplace Transform Characterization
    4.3 Independence
5 Stochastic Integral
6 Vague Topology on M(G)
7 Point Processes
    7.1 Simple Point Processes
    7.2 Enumeration of Points
    7.3 Generating Function
    7.4 Factorial Powers and Factorial Moment Measures

2. Basic Models and Operations

In this comprehensive lecture, we delve into the essential concepts and operations related to random measures and point processes. We start with an in-depth examination of Poisson point processes, a fundamental element in stochastic processes, focusing on their Laplace transforms and characterizations.

  1. Following this, we explore various operations on random measures and point processes. Key operations such as superposition, thinning, and marking are discussed in detail, providing you with the tools necessary to manipulate and analyze these stochastic models effectively.

  2. We then move on to the construction of new models, introducing Cox, Gibbs, and cluster point processes. These models offer a versatile framework for representing and studying intricate random phenomena, enhancing your ability to model real-world situations.

  3. The lecture also covers shot-noise processes, emphasizing their Laplace transforms, second-order moments, and U-statistics. These concepts are particularly relevant for applications in fields like telecommunications and finance.

  4. Finally, we address sigma-finite random measures, which broaden the scope of random measures beyond local finiteness. This discussion highlights the increased flexibility and applicability of sigma-finite measures in various stochastic modeling scenarios.

Contents:
1 Poisson Point Processes
    1.1 Laplace Transform
    1.2 Characterizations
2 Operations on Random Measures and Point Processes
    2.1 Superposition
    2.2 Thinning of Points
    2.3 Image of a Random Measure
    2.4 Independent Displacement of Points
    2.5 Independent Marking of Points
    2.6 Marked Random Measures
    2.7 Mixtures
3 Constructing New Models
    3.1 Cox Point Processes
    3.2 Gibbs Point Processes
    3.3 Cluster Point Processes
    3.4 Powers and Factorial Powers
4 Shot-Noise
    4.1 Laplace Transform
    4.2 Second Order Moments
    4.3 U-Statistics
5 Sigma-Finite Random Measures

3. Palm Theory

In this lecture we delve into the intricate and essential aspects of Palm theory, a cornerstone in the study of random measures and point processes. This lecture provides a comprehensive examination of Palm distributions, which are crucial for understanding the conditional properties of point processes given the presence of a point at a specific location.

  1. We begin with an in-depth analysis of Palm distributions, including the reduced Palm distribution, mixed Palm version, and local Palm probabilities. These foundational concepts are pivotal for characterizing the behavior and structure of point processes.
  2. The lecture then explores Palm distributions for specific models, such as Poisson point processes, Cox point processes, Gibbs point processes, and marked random measures. Each model is discussed in detail to illustrate the practical applications and theoretical implications of Palm theory in various contexts.
  3. Finally, we address higher-order Palm and reduced Palm distributions, extending the basic concepts to interactions involving multiple points.

Contents:
1 Palm Distributions
    1.1 Reduced Palm Distribution
    1.2 Mixed Palm Version
    1.3 Local Palm Probabilities
2 Palm Distributions for Particular Models
    2.1 Palm for Poisson Point Processes
    2.2 Palm for Cox Point Processes
    2.3 Palm Distribution of Gibbs Point Processes
    2.4 Palm Distribution for Marked Random Measures
3 Higher Order Palm and Reduced Palm
    3.1 Higher Order Palm
    3.2 Higher Order Reduced Palm

4. Transforms and Moment Measures

This lecture provides a rigorous introduction to transforms and moment measures as essential tools for characterizing random measures and point processes.

We begin with a detailed examination of characteristic functions and their associated cumulant and factorial cumulant measures, establishing the foundational concepts needed for subsequent analysis.

We then develop a range of both finite and infinite series expansions for the characteristic, Laplace, and generating functions, equipping attendees with a robust analytical framework.

Finally, we delve into the factorial moment expansion, highlighting its versatility through applications to marked point processes and shot-noise functions. This exploration reveals the intricate connections between the probabilistic nature of point processes and their analytical representations.

Contents:

  1. Characteristic Function
    1. Cumulant Measures
    2. Factorial Cumulant Measures
  2. Finite Series Transform Expansions
    1. Characteristic Function Expansion
    2. Laplace Transform Expansion
    3. Generating Function Expansion
  3. Infinite Series Transform Expansions
    1. Void Probability Expansion
    2. Symmetric Enumeration of Atoms of Finite Point Processes
    3. Janossy Measures
    4. Moment versus Janossy Measures
    5. Janossy versus Moment Measures
    6. Distribution of a Finite Point Process
    7. Order Statistics on R
  4. Factorial Moment Expansion
    1. Point Processes on R
    2. General Marked Point Processes
      1. Measurable Order
      2. Telescoping Formula
      3. Factorial Moment Expansion for Marked Point Processes
      4. Expansion Kernels
      5. Factorial Moment Expansion over Kernels
    3. Shot-Noise Functions

5. Determinantal and Permanental Point Processes

Access the detailed lecture notes in PDF format: Determinantal and Permanental Point Processes Lecture Notes
.

This lecture offers a comprehensive exploration of Determinantal and Permanental Point Processes (DPPs/PPPs), powerful tools for modeling spatial data with complex dependencies. We explore key aspects of these processes in a structured manner:

  1. Determinantal Point Process Basics: A precise formulation of DPPs, including their defining kernels and background measures, along with fundamental properties, such as
    thinning, simplicity, and restrictions, with illustrative examples including the Poisson process.
  2. Kernel Indistinguishability and Uniqueness: The concept of μ-indistinguishable kernels and the conditions that ensure a DPP/PPP is uniquely determined by its kernel and background measure.
  3. Generating Function and Laplace Transform: Derivation of expressions for the generating function and Laplace transform, essential for analyzing distributional properties.
  4. Existence with Regular Kernels: Definition of regular kernels and detailing general assumptions on the kernels for the existence of the determinantal point processes, and review examples of canonical point processes.
  5. α-Determinantal Point Processes: Introduction of a broader class of point processes including DPPs and PPPs and discuss conditions for their construction via superpositions.
  6. Palm Distributions: Derivation of The palm version of a Determinantal Point Processes
  7. Stationary Determinantal Point Processes: Discussion of Ginibre Determinantal Point Process & shift-invariant kernel
  8. Discrete Determinantal Point Processes: Detailed description of Characterization & Regularity.

The lecture content is organized as follows:

  • 1 Determinantal Point Process Basics
    • Definition and Basic Properties
    • Indistinguishable Kernels
    • Uniqueness of the Distribution
    • Generating Function and Laplace Transform
    • Inequalities for Moment Measures
  • 2 Existence of Determinantal Point Processes with Regular Kernels
    • Canonical Determinantal Point Processes
    • Integral Operator: Essentials
    • Canonical Version of a Kernel
    • Regular Kernels
  • 3 α-Determinantal Point Processes
    • Definition and Basic Properties
    • Uniqueness of the Distribution of an α-Determinantal Point Process
    • Generating Function and Laplace Transform
    • Permanental Point Process as Cox Point Process
    • Existence of α-Determinantal Point Processes
  • 4 Laplace Transform and Janossy Measures Revisited
    • Laplace Transform as Operator Determinant
    • Janossy Measures of α-Determinantal Point Processes; α ∈ {1/m : m ∈ N∗}
    • Janossy Measures of α-Determinantal Point Processes; α ∈ {−1/m : m ∈ N∗}
  • 5 Palm Distributions of Determinantal Point Processes
  • 6 Stationary Determinantal Point Processes on Rd
    • Ginibre Determinantal Point Process
    • Shift-invariant kernel
  • 7 Discrete Determinantal Point Processes
    • Characterization
    • Regularity
    • Janossy Measures
    • Palm Version

6. Palm Theory in the Stationary Framework

Access the detailed lecture notes in PDF format: Palm Theory in the Stationary Framework Lecture Notes.

This lecture focuses on Stationary Random Measures and Point Processes, emphasizing the significance of Palm Theory within the Stationary Framework. The content is structured to provide a comprehensive understanding of the foundational concepts and advanced applications of Palm probabilities.

  1. The first section introduces Palm Probabilities in the Stationary Framework, covering essential topics such as the Shift Operator, Flow, and Compatibility. We delve into the Palm Probability of a Random Measure, the Campbell-Little-Mecke-Matthes Theorem, and the Mass Transport Formula, culminating in Mecke’s Invariance Theorem.
  2. The second section presents the Palm Inversion Formula, exploring concepts like Voronoi Tessellation, the Inversion Formula, and the distinction between Typical and Zero Cells. We also discuss the Particular Case of the Line and Renewal Processes, followed by the Direct and Inverse Construction of Palm Theory.
  3. The final section examines Further Properties of Palm Probabilities, including Independence, Superposition, and Neveu’s Exchange Formula. We also cover the Alternative Version of Neveu’s Exchange Theorem, the Holroyd-Peres Representation of Palm Probability, and the Reduced Second Moment Measure.

The lecture content is organized as follows:

  1. Palm Probabilities in the Stationary Framework
    • Stationary Framework
    • Shift Operator and Stationarity
    • Flow and Compatibility
    • Palm Probability of a Random Measure
    • Campbell-Little-Mecke-Matthes Theorem
    • Mass Transport Formula
    • Mecke’s Invariance Theorem
  2. Palm Inversion Formula
    • Voronoi Tessellation
    • Inversion Formula
    • Typical versus Zero Cell
    • Particular Case of the Line
    • Renewal Processes
    • Direct and Inverse Construction of Palm Theory
    • Direct Construction: From Stationary to Palm Probability
    • Inverse Construction: From Palm to Stationary Probability
  3. Further Properties of Palm Probabilities
    • Independence
    • Superposition
    • Neveu’s Exchange Formula
    • Alternative Version of Neveu’s Exchange Theorem
    • Holroyd-Peres Representation of Palm Probability
    • Reduced Second Moment Measure

7. Marks in the Stationary Framework

Access the detailed lecture notes in PDF format: Marks in the Stationary Framework Lecture Notes.

This lecture, titled « Stationary Random Measures and Point Processes: Marks in the Stationary Framework, » explores the fascinating intersection of point processes, random measures, and Palm theory within a stationary framework. We delve into how to incorporate additional information, represented as « marks, » into these models.

  1. We begin by establishing the foundations of stationary marked random measures. This includes defining stationarity, introducing the shift operator, exploring compatibility with the flow, constructing stationary marked point processes, and considering the PASTA property in this more general setting.
  2. Next, we investigate marks that take values in general measurable spaces, a setting beyond the usual locally compact Hausdorff spaces. Key topics include examining marks generated by compatible stochastic processes, understanding the « shadowing property« , presenting a generalized Campbell-Little-Mecke-Matthes theorem applicable in this context, exploring mark-dependent thinning, and analyzing various transformations of stationary point processes based on their marks.
  3. Finally, we apply Palm theory to the specific case of stationary marked random measures. This involves analyzing the Palm distribution of the mark, exploring the Palm distributions of the marked random measures, and developing the concept of the Palm probability conditional on the mark to bridge the gap between the point process and the mark.

Table of Contents:

  • 1. Stationary Marked Random Measures
    • Stationary Marked Point Processes
    • Extension of PASTA to Rd
  • 2. Marks in a General Measurable Space
    • Selected Marks and Conditioning
    • Transformations of Stationary Point Processes Based on Marks
  • 3. Palm Theory for Stationary Marked Random Measures
    • Palm Distribution of the Mark
    • Palm Distributions of Marked Random Measures
    • Palm Probability Conditional on the Mark

Book: Point-processes, Random measures, and stochastic geometry

F. Baccelli, B. Błaszczyszyn, M.K. Karray (2020). Random measures, point processes and stochastic geometry. [ HAL ]

Our Work in the Scientific Community

I’m pleased that Paul Keeler (hpaulkeeler.com) references my point process resources at mohamedkadhem.com/point-processes, particularly highlighting the Palm calculus aspects related to telecommunication network modeling using stochastic geometry. This mention appears on this page of his website. It’s great to see this work getting visibility in the community!

Last Updated on 1 jour by Mohamed Kadhem KARRAY

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