Point Processes and Random Sets

A.Y. 2020/2021
Overall hours
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.
Course syllabus and organization

Single session

Lesson period
Second semester
Lessons will be held synchronously via the Zoom platform and / or MSTeams accordingly with the lessons timetable. Asynchronous lectures will be considered (via pre-recorded video) either in the case of technical problems or in the case of request by the students. In any case, a video recording of the lesson will be provided, available on the course reference site on the Ariel platform.
In cases of asynchronous lessons during the lessons timetable, the lecturer will be available via chat or videoconference to answer any questions by students.

Possible meetings in presence will be proposed.

The final examination will consist in an oral exam, via Zoom or MSTeams.
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. Compensator and stochastic intensity. Links with martingale theory.
2.7. Palm distributions (hint)
2.8. Superposition, thinning, clustering
3. Random closed sets
3.1. Definitions and examples
3.2. Capacity functional and the Choquet theorem
3.3. Random variables as particular 0-dimensional random sets
3.4. Discrete, continuous and absolutely continuous random closed sets
3.5. Weak convergence of random closed sets (hint)
3.6. Particle process and germ-grain-process
3.7. The Boolean model
3.8. Cluster Poisson process
3.9. Mean density of random closed sets (hint)
4. Some examples of application
4.1. Birth-and-growth processes
4.2. Fibre processes
4.3. Random tessellations
Prerequisites for admission
A basic course in Probability and Mathematical Statistics.
A basic course in Measure Theory and abstract integration
Teaching methods
Frontal lessons
Teaching Resources
Lecture notes will be provided as a guide to the study.

Additional bibliography :
1] Baddeley, A.; Bárány, I.; Schneider, R.; Weil, W. Stochastic Geometry. Lecture Notes in Math. 1892. Springer, Berlin, 2007.
2] Chiu, S. N.; Stoyan, D.; Kendall, W. S.; Mecke, J. Stochastic geometry and its applications - Third edition. John Wiley & sons, Chichester, 2013.
3] Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Vol. I. Elementary theory and methods. Springer-Verlag, New York, 2003.
4] Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Vol II. General theory and structure. Springer, New York, 2008.
5] Matheron, G. Random sets and integral geometry. John Wiley &Sons, New York-London-Sydney, 1975.
6] Molchanov, I. Theory of random sets. Probability and its Applications. Springer-Verlag London, Ltd., London, 2005.
7] Schneider, R.; Weil, W. Stochastic and integral geometry. Springer-Verlag, Berlin, 2008.
8] O.E. Barndorff-Nielsen, W.S. Kendall, M.N.M. van Lieshout Editors. Stochastic Geometry. Likelihood and Computation, Chapman & Hall/CRC, 1999.
9] E. Spodarev Editor. Stochastic Geometry, Spatial Statistics and Random Fields. Asymptotic methods. Springer- Verlag, 2013
Assessment methods and Criteria
The final examination consists of an oral exam.
In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The final examination is passed if the oral exam is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Lessons: 42 hours
Professor: Villa Elena
Educational website(s)