Point Processes, Random Measures, and Stochastic Geometry: Lectures

Point-Processes

Course Overview

These lecture notes are based on the book « Random Measures, Point Processes and Stochastic Geometry » (F. Baccelli, B. Błaszczyszyn, M.K. Karray). These materials explore random structures embedded in Euclidean or more general topological spaces. The primary focus is on random measures, point processes, and stochastic geometry, which have become crucial tools in diverse fields, including natural sciences (cosmology, ecology, cell biology), engineering (material sciences, networks), and emerging fields like data science.

These notes, derived from the book, are designed to provide a clear and accessible path from fundamental definitions and properties to the application of these mathematical objects in concrete stochastic models. The book offers a rigorous and complete development of the theory, with all proofs included and special attention paid to measurability issues. A key strength of the book, and reflected in these notes, is the inclusion of elaborate examples (« models ») drawn from various fields, translated into clear mathematical terms, bridging theory and application. The material covers both structural results on stationary random measures and stochastic geometry objects and computational results on important parametric classes of point processes (especially Poisson and Determinantal).

The structure of the book is mirrored in these notes: Part I covers Foundations, Part II focuses on the Stationary Framework, and Part III explores Stochastic Geometry. To ensure a rigorous and self-contained treatment, the book provides in the appendix essential background material.

These lecture notes assume a basic familiarity with measure and probability theory. In these notes, we aim to provide clarifications, elaborations, additional examples, and guidance on applying these powerful tools to your own research and projects. These materials are foundational for the analysis of Queueing Theory and Wireless Networks, where point processes serve as core modeling tools. We hope you find these lecture notes helpful in your journey to master random measures, point processes, and stochastic geometry!

I. Random Measures and Point Processes

1 Foundations

PDF of the lecture on Foundations of random measures and point processes, an in-depth exploration of these foundational stochastic objects.
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We begin by establishing the necessary framework, including key structures such as mean measures, Laplace transforms, void probabilities, and Campbell’s averaging formula. This foundational section sets the stage for understanding the fundamental properties and behaviors of random measures and point processes. We then delve into the characterization of distributions, exploring powers and moment measures, the role of the Laplace transform in characterizing distributions, and the concept of independence in the context of random measures. We introduce stochastic integrals with respect to random measures and discuss the vague topology on the space of measures. Finally, we provide a detailed examination of point processes, including simple point processes, methods for enumerating points, the generating function, and factorial powers and factorial moment measures crucial for higher-order properties.

Course Outline:

  1. Framework
    1. Basic Spaces and Structures
    2. Mean Measure, Laplace Transform and Void Probability
    3. Campbell’s Averaging Formula
    4. Powers and Moment Measures
    5. Polish and l.c.s.h. Spaces
  2. Distribution Characterization
    1. Characterization via Generating Subclass
    2. Independence
    3. Laplace Transform Characterization
  3. Stochastic Integral
  4. Point Processes
    1. Point Measures and Processes
    2. Generating Function
    3. Factorial Powers and Factorial Moment Measures
    4. Counting Measures Point of View
    5. Measurable Enumeration of Points
  5. Vague Topology on M̄(G)

2 Basic Models and Operations

PDF of the lecture on Basic Models and Operations for random measures and point processes.
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We begin with an in-depth examination of Poisson point processes, a fundamental element of stochastic processes, focusing on their Laplace transforms and characterizations. We then explore key operations on random measures and point processes such as superposition, thinning, image, independent displacement, independent marking, and mixtures. We move on to the construction of new models, introducing Cox, Gibbs, and cluster point processes, which offer a versatile framework for representing complex random phenomena. We address shot-noise processes, emphasizing their Laplace transforms, second-order moments, and U-statistics, particularly relevant for telecommunications and finance. Finally, we discuss sigma-finite random measures, broadening the scope of random measures beyond local finiteness.

Course Outline:

  1. Poisson Point Processes
    1. Laplace Transform
    2. Characterizations
  2. Operations on Random Measures and Point Processes
    1. Superposition
    2. Thinning of Points
    3. Image of a Random Measure
    4. Independent Displacement of Points
    5. Independent Marking of Points
    6. Marked Random Measures
    7. Mixtures
  3. Constructing New Models
    1. Cox Point Processes
    2. Gibbs Point Processes
    3. Cluster Point Processes
    4. Powers and Factorial Powers
  4. Shot-Noise
    1. Laplace Transform
    2. Second Order Moments
    3. U-Statistics
  5. Sigma-Finite Random Measures

3 Palm Theory

PDF of the lecture on Palm Theory, a cornerstone in the study of random measures and point processes.
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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. We then explore Palm distributions for particular models such as Poisson point processes, Cox point processes, Gibbs point processes, and marked random measures, illustrating the practical applications and theoretical implications of Palm theory. Finally, we address higher-order Palm and reduced Palm distributions, extending the basic concepts to interactions involving multiple points.

Course Outline:

  1. Palm Distributions
    1. Reduced Palm Distribution
    2. Mixed Palm Version
    3. Local Palm Probabilities
  2. Palm Distributions for Particular Models
    1. Palm for Poisson Point Processes
    2. Palm for Cox Point Processes
    3. Palm Distribution of Gibbs Point Processes
    4. Palm Distribution for Marked Random Measures
  3. Higher Order Palm and Reduced Palm
    1. Higher Order Palm
    2. Higher Order Reduced Palm

4 Transforms and Moment Measures

PDF of the lecture on Transforms and Moment Measures, essential tools for characterizing random measures and point processes.
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We begin with a detailed examination of characteristic functions and their associated cumulant and factorial cumulant measures. We then develop a range of finite series expansions for the characteristic, Laplace, and generating functions, equipping a robust analytical framework. We extend this to infinite series transform expansions, including void probability expansion, symmetric enumeration of atoms of finite point processes, Janossy measures, the relationship between moment and Janossy measures, the distribution of a finite point process, and order statistics on R. Finally, we delve into the factorial moment expansion, highlighting its versatility through applications to marked point processes, with the measurable order, the telescoping formula, the factorial moment expansion for marked point processes, expansion kernels, the factorial moment expansion over kernels, and applications to shot-noise functions.

Course Outline:

  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
    3. Shot-Noise Functions

5 Determinantal and Permanental Point Processes

PDF of the lecture on Determinantal and Permanental Point Processes (DPPs/PPPs), powerful tools for modeling spatial data with complex dependencies.
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We begin with determinantal point process basics, a precise formulation including defining kernels and background measures, fundamental properties such as thinning, simplicity, and restrictions, the concept of μ-indistinguishable kernels, and the uniqueness of the distribution. We derive expressions for the generating function and Laplace transform, essential for analyzing distributional properties, along with inequalities for moment measures. We then examine existence of determinantal point processes with regular kernels, including canonical determinantal point processes, integral operator essentials, and the canonical version of a kernel. We introduce α-determinantal point processes, a broader class including DPPs and PPPs, and discuss their construction via superpositions. We revisit the Laplace transform and Janossy measures through operator determinants. Finally, we derive Palm distributions of determinantal point processes, study stationary determinantal point processes on Rd including the Ginibre determinantal point process and shift-invariant kernels, and conclude with discrete determinantal point processes.

Course Outline:

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

II. Stationary Random Measures and Point Processes

6 Palm Theory in the Stationary Framework

PDF of the lecture on Palm Theory in the Stationary Framework, emphasizing the significance of Palm probabilities in the stationary case.
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We begin with Palm probabilities in the stationary framework, covering the shift operator and stationarity, flow and compatibility, the Palm probability of a random measure, the Campbell-Little-Mecke-Matthes theorem, the mass transport formula, and culminating in Mecke’s invariance theorem. We then develop the Palm inversion formula, exploring the Voronoi tessellation, the inversion formula itself, the distinction between typical and zero cells, the particular case of the line, renewal processes, and the direct and inverse construction of Palm theory. Finally, we examine further properties of Palm probabilities, including independence, superposition, Neveu’s exchange formula, the Holroyd-Peres representation of Palm probability, and the reduced second moment measure.

Course Outline:

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

7 Marks in the Stationary Framework

PDF of the lecture on Marks in the Stationary Framework, exploring how to incorporate additional information as « marks » into stationary models.
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We begin by establishing the foundations of stationary marked random measures, including defining stationarity, the shift operator, compatibility with the flow, constructing stationary marked point processes, and the PASTA property in this general setting. We then investigate marks in a general measurable space, beyond locally compact Hausdorff spaces, examining marks generated by compatible stochastic processes, the shadowing property, a generalized Campbell-Little-Mecke-Matthes theorem, mark-dependent thinning, and various transformations of stationary point processes based on marks. Finally, we apply Palm theory to stationary marked random measures, analyzing the Palm distribution of the mark, the Palm distributions of marked random measures, and the Palm probability conditional on the mark.

Course Outline:

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

8 Ergodicity

PDF of the lecture on Ergodicity for stationary random measures and point processes.
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We begin with the necessary background and motivation for studying ergodicity in the context of stationary random measures and point processes. We then review Birkhoff’s pointwise ergodic theorem, a cornerstone of ergodic theory essential for the subsequent developments. We then develop ergodic theorems for random measures, introducing the concept of ergodicity of random measures, demonstrating the ergodic theorem for random measures, and covering cross-ergodicity. Finally, we explore the ergodicity of marked random measures, extending the ergodic framework to processes where each point carries additional information.

Course Outline:

  1. Motivation
  2. Birkhoff’s Pointwise Ergodic Theorem
  3. Ergodic Theorems for Random Measures
    1. Ergodicity of Random Measures
    2. Ergodic Theorem for Random Measures
    3. Cross-Ergodicity
  4. Ergodicity of Marked Random Measures

III. Stochastic Geometry

9 Framework for Stochastic Geometry

PDF of the lecture on the Framework for Stochastic Geometry, providing the mathematical tools underpinning the analysis of spatial random structures.
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We begin by establishing a rigorous foundation in the space of closed sets, essential for defining and characterizing random geometric structures. We then introduce the core concept of a random closed set (RCS), focusing on its characterization through the powerful capacity functional. We explore set processes and their connection to RCSs, leading to a detailed analysis of stationary RCSs and set processes. We devote a significant portion to understanding the fundamental characteristics of random closed sets, including the capacity functional, coverage functions, and their properties. Finally, we focus on the specific characteristics of stationary random closed sets such as volume fraction, reduced covariance function, and contact distribution function, providing the necessary framework for analyzing spatial patterns and dependencies in stochastic geometric models.

Course Outline:

  1. Space of Closed Sets
  2. Random Closed Sets (RCS)
    1. The Capacity Functional
    2. Set Processes
    3. Stationary RCS and Set Process
    4. Characteristics of Random Closed Set
    5. Characteristics of Stationary Random Closed Set

10 Coverage and Germ-Grain Models

PDF of the lecture on Coverage and Germ-Grain Models, fundamental tools for understanding irregular geometric patterns.
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We begin by formally defining coverage models as unions of random sets and establish key properties, focusing on their capacity functionals. We then delve into germ-grain models, which are constructed from point processes marked by some random sets called grains. We present a thorough analysis of germ-grain constructions, including stationarity conditions and derivations of mean measures. We discuss an inverse construction, which allows for the derivation of stationary marked point processes from stationary set processes. Finally, we showcase concrete examples of specialized coverage models, including hard-core coverage models (with non-overlapping grains) and shot-noise coverage models.

Course Outline:

  1. Coverage Model
  2. Germ-Grain Model
    1. Germ-Grain Construction
    2. Inverse Construction
  3. Further Examples
    1. Hard-Core Coverage Models
    2. Shot-Noise Coverage Models

11 Line Processes and Tessellations

PDF of the lecture on Line Processes and Tessellations, fundamental concepts in stochastic geometry.
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We begin with line processes in R², formally defining line processes in the two-dimensional Euclidean space. We introduce a bijective parameterization of lines using polar coordinates, then discuss stationary line processes, characterizing the conditions under which the underlying point process is invariant under cylinder shears. We cover the construction of line measures and their properties. We then shift to planar tessellations, partitions of the plane into cells. We explore two key examples: the Voronoi tessellation, constructed from a point process, and the Crofton tessellation, generated by a line process. We investigate their properties and the relationships between the point processes associated with their vertices, edges, and cells.

Course Outline:

  1. Line Processes in R²
    1. Parameterization of Lines in R²
    2. Stationary Line Processes
    3. Associated Random Measures
  2. Planar Tessellations
    1. Voronoi Tessellation
    2. Crofton Tessellation

12 Complements

PDF of the lecture on Complements, exploring a key property of Poisson point processes on l.c.s.h. spaces: the Strong Markov Property.
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We provide a formal definition of stopping sets with respect to a point process. We then present the strong Markov property for Poisson point processes on l.c.s.h. spaces, explaining how it extends the usual Markov property to cases where conditioning occurs on a random stopping set. Finally, we illustrate the application of this property by analyzing the distances from a fixed point to ordered points.

Course Outline:

  1. Strong Markov Property of Poisson Point Process

IV. Appendix

13 Transforms of Random Variables

PDF of the lecture on Transforms of Random Variables, comprehensive treatment of transforms used to characterize random variables and random vectors.
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We begin by examining random variables, exploring the theoretical aspects and practical applications of the characteristic function and generating function. We emphasize the relationships of these transforms with the moments and cumulants of a distribution, covering differentiability, Taylor expansions, and uniqueness theorems, with the interplay between ordinary moments and factorial moments. We then focus on nonnegative random variables, where the Laplace transform is introduced as a central analytical technique, with its properties, relationship to moments, and an alternative cumulant definition. Finally, we extend the concepts to random vectors, defining multivariate characteristic functions and generating functions and their relationship to mixed moments, then to nonnegative random vectors with the multivariate Laplace transform.

Course Outline:

  1. Random Variables
    1. Characteristic Function
    2. Generating Function
    3. Moments Versus Factorial Moments
    4. Ordinary Cumulants
    5. Factorial Cumulants
  2. Nonnegative Random Variables
    1. Laplace Transform
    2. Cumulants via the Laplace Transform
  3. Random Vectors
    1. Moments from Transforms
    2. Cumulants
    3. Nonnegative Random Vectors

14 Useful Results in Measure Theory

PDF of the lecture on Useful Results in Measure Theory, a comprehensive compendium of essential results serving as a foundational resource for advanced studies.
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We cover basic results including properties of negligible sets in product spaces, the inclusion-exclusion principle, and basic facts about symmetric measures. We delve into the concept of the support of a measure, characterizing the region where the measure is concentrated. We present the functional monotone class theorems for nonnegative functions, a powerful tool. We provide a comprehensive treatment of mixture and disintegration of measures, including measure kernels, probability kernels, the measure mixture theorem, and the measure disintegration theorem. We introduce power and factorial powers of measures. We establish equality of measures on product spaces. Finally, we introduce symmetric complex Gaussian random variables and vectors with Wick’s formula.

Course Outline:

  1. Basic Results
  2. Support of a Measure
  3. Functional Monotone Class Theorems
  4. Mixture and Disintegration of Measures
    1. Mixture of Measures
    2. Disintegration of Measures
  5. Power and Factorial Powers of Measures
  6. Equality of Measures
  7. Symmetric Complex Gaussian Random Variables

15 Useful Results in Algebra

PDF of the lecture on Useful Results in Algebra, a curated collection of algebraic results essential for advanced analysis.
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We dedicate the first section to matrices, delving into fundamental properties, determinants, and related inequalities, with a strong emphasis on complex matrices. Key topics include important inequalities like Hadamard’s inequality, Fischer’s inequality, and Koteljanskii’s inequality for Hermitian nonnegative-definite matrices. We provide an in-depth introduction to the Schur complement, including its formula, inverse, and applications. We cover various formulas for the diagonal expansion of the determinant, and introduce the α-determinant, a generalization of the standard determinant encompassing both the determinant and the permanent. The second section focuses on the composition of power series, establishing conditions for the validity of substituting one power series into another, with a crucial lemma on existence and convergence.

Course Outline:

  1. Matrices
    1. Inequalities
    2. Schur Complement
    3. Diagonal Expansion of the Determinant
    4. α-Determinant of a Matrix
  2. Power Series Composition

16 Useful Results in Functional Analysis

PDF of the lecture on Useful Results in Functional Analysis, focusing particularly on integral operators and the Fredholm determinant.
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We begin a study of integral operators, a distinguished class of linear operators acting on L² spaces, defined via integration against a kernel. We review the requisite properties of L² spaces, then explore key properties of integral operators including boundedness, compactness, the nature of their adjoints, and an analysis of integral operators with Hermitian kernels. We then introduce the Fredholm determinant, a sophisticated generalization of the determinant concept to the realm of trace class operators on separable Hilbert spaces. We carefully develop the Fredholm determinant using the framework of antisymmetric tensor products, culminating in Lidskii’s theorem, establishing a fundamental connection between the Fredholm determinant and the eigenvalues of the operator.

Course Outline:

  1. Integral Operators
    1. L² Space Properties
    2. Linear Operators
    3. Integral Operator Basics
    4. Trace Class Operators
    5. Hilbert-Schmidt Operators
    6. Restriction of Integral Operators
    7. Canonical and Pre-Canonical Kernels
    8. Continuous Kernels
    9. Nonnegative-Definite Property
  2. Operator Determinant
    1. Fredholm Determinant
    2. Expansion of Integral Operator’s Determinant

Book: Preliminary Version Available

The lectures on this page are based on the first preliminary version of our book, available as a preprint on the HAL open archive: F. Baccelli, B. Błaszczyszyn, M. K. Karray — Random Measures, Point Processes and Stochastic Geometry. Preliminary Version (Preprint). HAL open archive, 2024.

Underlying Space Properties

In the study of random measures and point processes, the choice of the underlying space plays a crucial role in determining the technical complexity and generality of the results. Many authors focus on Polish spaces (e.g., DaleyVereJones2003, DaleyVereJones2008, Kallenberg2017), while others work under the more restrictive assumption that the space is locally compact, second countable, and Hausdorff (l.c.s.h.) (e.g., Neveu1977, Kallenberg1983), which simplifies certain proofs and formulations.

In this book, we begin with general measurable spaces as the underlying framework, an approach also employed, in part, by Bremaud2020. When additional topological structure is needed, we focus on l.c.s.h. spaces. The l.c.s.h. framework encompasses a broad class of practically relevant spaces, including Euclidean spaces and more intricate examples, such as the space of nonempty closed subsets of a Euclidean space.

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!

Acknowledgements

The authors thank Oliver Diaz-Espinosa, Mayank Manjrekar, James Murphy, Eliza O’Reilly, Pierre Bernhard, Philippe Sarotte, and Ottmar Cronie for their comments on early versions of this manuscript.

About These Topics

These graduate-level lectures on point processes, random measures, and stochastic geometry are designed for students and researchers seeking a rigorous mathematical treatment of spatial random structures. The material develops the theory from foundational definitions through to advanced topics in stationary frameworks and geometric models, with applications to telecommunications, wireless networks, and statistical physics. Part I covers the foundations of random measures and point processes, including Laplace transforms, void probabilities, moment measures, and the fundamental Palm theory. It also covers the key parametric classes of Poisson point processes, Cox processes, Gibbs processes, cluster processes, shot-noise processes, and the family of determinantal and permanental point processes. Part II develops the stationary framework, including Palm theory in the stationary case with the Campbell-Little-Mecke-Matthes theorem, Mecke’s invariance theorem, the mass transport formula, the Palm inversion formula, the Voronoi tessellation, and Neveu’s exchange formula. Marks in the stationary framework and ergodic theorems are treated rigorously, building on Birkhoff’s pointwise ergodic theorem. Part III explores stochastic geometry, including random closed sets, the capacity functional, coverage and germ-grain models, line processes, and planar tessellations such as the Voronoi and Crofton tessellations. The strong Markov property of Poisson point processes on l.c.s.h. spaces is also developed. The Appendix provides essential background results in measure theory, algebra, and functional analysis, including a rigorous treatment of integral operators, trace class operators, and the Fredholm determinant.

Last Updated on 24 mai 2026 by Mohamed Kadhem KARRAY

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