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

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

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

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

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