Probability Theory II
Winter 2025/26
| Time: |
Di, Do 10-12
|
| Room:
|
05-136
|
| Begin: |
28.10.2025 |
Exam:
|
in Summer 2026 for the "vertiefungsmodul"
|
This lecture is aimed at students of mathematics.
It is the third part of a three-semester course. Together with the first two parts (Introduction to Probability and Statistics, Probability Theory I), it provides the knowledge of probability theory that every student should have for a diploma or master's degree in mathematics. Formally, it is the first part of the advanced module STO-002, which continues in the following semester with an advanced lecture in probability theory and an optional advanced seminar. Probability theory deals with the quantitative study of all phenomena in which chance plays a role. In Fermat's time, this mainly concerned games of chance – today, questions from statistical physics, biology, financial mathematics, statistics, and so on have moved to the forefront.
The first part comprises basic knowledge of probability theory and statistics, within a framework essentially free of measure theory. In the second part, the apparatus of measure theory was developed, essentially to the extent that it is necessary for probability theory.
In Probability Theory II, martingales are introduced and systematically examined, with particular attention being given to optional stopping theorems and martingale convergence theorems. We develop the theory of characteristic functions (Fourier transformation), thus deriving the central limit theorem in Lindeberg's form, and consider infinitely divisible and stable distributions. Using the construction of product spaces and the theorem on projective limits, we construct stochastic processes with a general time set. As an application of this general method, we learn the basic concepts of ergodic theory.
Literature
- Bauer: Maß- und
Integrationstheorie, de Gruyter
- Billingsley:
Probability
- Breiman:
Probability
- Durrett: Probability: Theory and
Examples
- Elstrodt:
Maß- und Integrationstheorie
- Felller, An introduction to
probability theory and its applications, Band 1 und Band 2.
- Georgii:
Stochastik
- Karatzas und Shreve: Brownian motion
and stochastic calculus
- Keller: Wahrscheinlichkeitstheorie,
Vorlesungsskript Erlangen.
- Klenke: Wahrscheinlichkeitstheorie,
Springer Verlag, 4. Auflage, 2020.
- Shiryaev:
Probability
a.k. 04.08.2025
