# Stochastic Geometry II

## Time and Place

Lecture

Wednesday, 10:15 - 11:55, E03, Heho 22

Exercise classes

Friday, 12:15 - 13:55, 120, Heho18

## Type

2 hours lecture, 1 hour exercise classes

4 credit points

## Prerequisites

Basic knowledge of probability like taught in "Elementare Wahrscheinlichkeitsrechnung und Statistik". The lecture "Stochastic Geometry" is not necessary.

## Content

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.

### Random tessellations

• 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.

## Exam

There is no prerequisite. Further details will be announced later.

## Exercise sheets

Submitting solutions for the exercise sheets is strongly recommanded.

## Literature

• 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.

# Contact

Lecturer & Teaching assistant

Helmholtzstraße 18, Raum 1.46

0731 50 23526

juergen.kampf(at)uni-ulm.de

# News

The language of the course has been changed to german.

Only the german version of this page will be updated further.