Universität Duisburg-Essen

Fakultät für Mathematik, Campus Essen

Arbeitsgruppe Wahrscheinlichkeitstheorie

Wolfgang Löhr
## Lecture (3+1SWS), Winter 2016/17, "Vertiefungsmodul" for master students

# Large Deviations (Große Abweichungen)

**Time and Room:** Tuesday 10:15h - 12:00h in WSC-S-U-3.02 (**new room WSC-S-3.14**) and Thursday, 10:15h - 12:00h in WSC-S-U-3.01
**No class** from 21st of November until 3rd of December!
**ECTS credits:** 6

If you need 9 ECTS credits, it is possible to give a presentation at the end of the semester.
The course is directed to master students (Vertiefungsmodul Stochastik). It integrates lectures and exercise
classes of 4SWS in total, roughly split as 3SWS of lecture and 1SWS of exercises.
The course has a final exam (written or oral depending on number of participants). The
instruction language is English, unless all participants prefer German.
**Content of the Lecture:**
- Cramér Theorem
- Large deviation principle on Polish spaces
- Exponential tightness
- Varadhan's Lemma
- Bryc's Formula
- Gärtner-Ellis Theorem
- Sanov Theorem
- Schilder and Strassen Theorems

Large deviation theory deals with "rare events" (with exponentially low probability, such as geting head more
than 60% of the time when throwing a fair coin very often) and the quantification of how rare they are in terms
of an exponential decay rate. Thus one obtains the speed of convergence for laws of large numbers.
We develop an abstract theory in analogy to the theory of weak convergence on Polish spaces.
Large deviation theory has many applications in information theory, ergodic theory, statistics, statistical
mechanics, financial mathematics and other fields.

**Prerequisites:**

- Probability Theory I
- Probability Theory II is recommended

**Literature:**
- Amir Dembo, Ofer Zeitouni: Large Deviations Techniques and Applications, second edition, Springer (1998)

Online-Zugriff via the university library
- Wolfgang König's lecture notes (German): pdf
- Jin Feng, Thomas Kurtz: Large Deviations for Stochastic Processes, AMS (2006)
- Chapter 27 of Kallenberg: Foundations of Modern Probability, second edition, Springer (2001)
- Chapter 23 of Achim Klenke: Wahrscheinlichkeitstheorie, Springer (2006)

**Exercise sheets:**

Homepage von Wolfgang Löhr

Last updated: 2016/11/17