Universität Wien

250189 VO Advanced Probability Theory (2023S)

7.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Thursday 02.03. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 06.03. 11:30 - 13:00 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 09.03. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 16.03. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 20.03. 11:30 - 13:00 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 23.03. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 27.03. 11:30 - 13:00 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 30.03. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 17.04. 11:30 - 13:00 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 20.04. 09:45 - 11:15 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Monday 24.04. 11:30 - 13:00 Seminarraum 5 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 27.04. 09:45 - 11:15 Seminarraum 5 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 04.05. 09:45 - 11:15 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 08.05. 11:30 - 13:00 Seminarraum 4 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 11.05. 09:45 - 11:15 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 15.05. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 22.05. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 25.05. 09:45 - 11:15 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 01.06. 09:45 - 11:15 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 05.06. 11:30 - 13:00 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 12.06. 11:30 - 13:00 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 15.06. 09:45 - 11:15 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 19.06. 11:30 - 13:00 Seminarraum 4 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 22.06. 09:45 - 11:15 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 26.06. 11:30 - 13:00 Seminarraum 6 Oskar-Morgenstern-Platz 1 1.Stock

Information

Aims, contents and method of the course

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.

Assessment and permitted materials

The course is assessed based on performance in a written exam at the end of the course or after the course. Oral exams may be organized for those who wish to retake the exam.

Minimum requirements and assessment criteria

To pass the course, the student is required to gain a basic understanding of measure-theoretic probability and to be able to tackle simple common applications of the theory. For a high grade, a good command of the more advanced topics and an ability to apply them in various examples is required. For grade 4, around 50% of the maximum points of the exam will be required.

There are no formal prerequisites for this course. However, some basic measure theory (eg. some of the core contents in the course "Measure and integration theory"), as well as its prerequisites, are necessary to understand the contents of this course. These prerequisites will be quickly reviewed at the beginning of the course, and a student not familiar with measure theory is advised to invest a fair amount of time to study these along the course. Basic skills in probability calculus are very useful, although not formally required.

Examination topics

The exam is based on the lecture material of the course. Knowing percolation theory is not formally required in the exam, but many tools involved in it and belonging to the core course material may be asked. Solving exercise problems and participating in the Introductory Seminar (i.e. the exercise class) is very helpful for preparing for the exam.

Reading list

There will be lecture notes, which will be updated along the lectures. Some potentially useful references and materials for further study are the following.

Books:
- P. Billingsley: Probability and measure ( https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf )
- R. Durrett: Probability: theory and examples ( https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf )
- D. Williams: Probability with martingales
- G. Grimmett and D. Stirzaker: Probability and Random Processes

Lecture notes:
- G. Miermont: Advanced probability ( http://perso.ens-lyon.fr/gregory.miermont/AdPr2006.pdf )
- K. Izyurov: Probability theory ( https://wiki.helsinki.fi/display/mathphys/Izyurov?preview=/123044553/213983389/Notes_28.11.pdf )

Association in the course directory

MSTW

Last modified: We 06.03.2024 12:26