Stochastic Geometry II
Time and Place
Wednesday, 10:15 - 11:55, E03, Heho 22
Friday, 12:15 - 13:55, 120, Heho18
2 hours lecture, 1 hour exercise classes
4 credit points
Basic knowledge of probability like taught in "Elementare Wahrscheinlichkeitsrechnung und Statistik". The lecture "Stochastic Geometry" is not necessary.
Stochastic geometry deals with random geometric structures. In this lecture we will treat two selected topics from stochastic geometry:
Random compact sets
- The convex hull of random points: We consider the convex hull of finitely many points distributed uniformly within a given compact set (left picture). We will examine stochastic properties (e.g. the expected value) of geometric quantities (e.g. the area) of this convex hull.
- The strong law of large numbers: The arithmetic mean convergences to the expected value for random compact sets as well if we define these notions appropriately.
- Definition and elementary properties: Tessellations partition the Euclidean space into so-called cells.
- Mean values for geometric quantities of a cell: We will define the notion of the typical distribution of a cell and derive formulas for the expected values of various geometric quantities, e.g. the area or the number of neighboring cells.
- Models for random tessellations: We will treat the most important models of random tessellations: Voronoi tessellation (middle picture), hyperplane tessellation (right picture) and STIT tessellation.
There is no prerequisite. Further details will be announced later.
Submitting solutions for the exercise sheets is strongly recommanded.
- Schneider, R. and Weil, W.: Stochastic and Integral Geometry. Springer, 2008.
- Molchanov, I.: Theory of Random Sets. Springer, 2005.
- (Chiu, S.,) Stoyan, D., Kendall, W. and Mecke, J.: Stochastic Geometry and Its Applications. Wiley, 1995, 2008, 2013.
- Spodarev, E.: Stochastic Geometry, Spatial Statistics and Random Fields. Springer, 2013.
- Schmidt, V..: Stochastic Geometry, Spatial Statistics and Random Fields. Springer, 2015.