250189 VO Advanced Probability Theory (2023S)

This course focuses on modern probability theory in its measure-theoretic framework. Its aim is to provide students with a deeper understanding of randomness and to introduce tools to tackle various applications. The course also forms a solid basis for more specialized courses in the Stochastics curriculum. As a beautiful application of some of the core contents, we also introduce percolation theory, a modern "subfield" of probability theory, which could be considered as a part of common knowledge of probabilists (and where three Fields medals have been awarded).

The core contents of the course include:
- definition of probability space and basic notions of measure-theoretic probability
- random variables, expectation, independence
- Borel-Cantelli lemmas, Kolmogorov zero-one law
- law of large numbers
- notions of convergence, such as convergence in probability and weak convergence
- central limit theorem
- conditional expectations
- martingales
- optional stopping

The method of the course is following the lectures and taking a final exam. Attendance in the lectures is strongly recommended since they include all the exam contents as well as enable mutual interaction to provide better understanding. In addition, it is strongly recommended to solve exercise problems and participate in the exercise classes, which comprise the course "Introductory Seminar on Advanced Probability Theory" ( https://ufind.univie.ac.at/en/course.html?lv=250185&semester=2023S ). The exercises are evaluated separately as part of the "Introductory Seminar" and do not contribute to the grade of this lecture course.