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, also taking care of the related statistical techniques.
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.
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
4. Some examples of application 4.1. Birth-and-growth processes 4.2. Fibre processes 4.3. Random tessellations