Point Processes and Random Sets
A.Y. 2025/2026
Learning objectives
The main target of the course is to provide the basics of the theory of random closed sets and of spatial point processes, which are often used to model many real phenomena in applications. Some examples of applications of such random geometrical processes will be discussed in more detail.
Expected learning outcomes
Basics in the Theory of Point Processes and in Stochastic Geometry. The student will then be able to apply and broaden his/her knowledge of the subjects in different areas of interest, both in theoretical and applied contexts.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Introduction
1.1. Random closed sets and point processes: general ideas
1.2. Some fields of application
2. Point processes
2.1. Basic properties and definitions
2.2. Intensity measure and moment measures
2.3. Main point processes
2.4. Marked point process
2.5. Poisson marked point process
2.6. Palm distributions
2.7. Principal operations on point processes
3. Point processes on the real line.
3.1. Compensator and stochastic intensity.
3.2. Stochastic integral with respect to a point process.
3.3. Links with martingale theory
4. Random closed sets
4.1. Definitions and examples
4.2. Capacity functional and the Choquet theorem
4.3. Particle process and germ-grain-process
4.4. The Boolean model
4.5. Some problems of applicative interest
1.1. Random closed sets and point processes: general ideas
1.2. Some fields of application
2. Point processes
2.1. Basic properties and definitions
2.2. Intensity measure and moment measures
2.3. Main point processes
2.4. Marked point process
2.5. Poisson marked point process
2.6. Palm distributions
2.7. Principal operations on point processes
3. Point processes on the real line.
3.1. Compensator and stochastic intensity.
3.2. Stochastic integral with respect to a point process.
3.3. Links with martingale theory
4. Random closed sets
4.1. Definitions and examples
4.2. Capacity functional and the Choquet theorem
4.3. Particle process and germ-grain-process
4.4. The Boolean model
4.5. Some problems of applicative interest
Prerequisites for admission
A basic course in Probability
A basic course in Measure Theory and abstract integration
A basic course in Measure Theory and abstract integration
Teaching methods
Lectures
Teaching Resources
Principal bibliography:
1] Baccelli F., Blaszczyszyn B., Karray M., Random Measures, Point Processes, and Stochastic Geometry. Inria, 2020. hal-02460214
2] Chiu, S., Stoyan D., Kendall W.S., Mecke J., Stochastic Geometry and its Application- Third edition, John Wiley & sons, Chichester, 2013.
3] Brémaud, P.: Point process calculus in time and space - an introduction with applications, Springer, Cham, 2020.
Lecture notes will be provided as a study guide, along with additional bibliographic references.
1] Baccelli F., Blaszczyszyn B., Karray M., Random Measures, Point Processes, and Stochastic Geometry. Inria, 2020. hal-02460214
2] Chiu, S., Stoyan D., Kendall W.S., Mecke J., Stochastic Geometry and its Application- Third edition, John Wiley & sons, Chichester, 2013.
3] Brémaud, P.: Point process calculus in time and space - an introduction with applications, Springer, Cham, 2020.
Lecture notes will be provided as a study guide, along with additional bibliographic references.
Assessment methods and Criteria
The final examinationconsists of an oral test.
During the oral test, students will be asked to illustrate some results from the course syllabus in order to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if the oral test is passed; the grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test
During the oral test, students will be asked to illustrate some results from the course syllabus in order to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if the oral test is passed; the grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professors:
Fuhrman Marco Alessandro, Villa Elena
Professor(s)
Reception:
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.