Course "Large Deviations", TU Berlin, Summer Semester 2020
Basic info
- Lecturer: Michiel Renger
- Weekly hourse (SWS): 2.0
- Material:
- Wolfgang König - Große Abweichungen, Techniken und Anwendungen (lecture notes) For those who do not speak German; most material can also be found in the books by Dembo & Zeitouni, or the book by den Hollander
- Amir Dembo and Ofer Zeituni - Large Deviations, Techniques and Applications (Springer). The TU Berlin has made this book digitally available for you through this link (should work from Campusnetz or VPN).
- Frank den Hollander - Large Deviations (AMS). The TU Berlin has made this book digitally available for you through this link.
- Language: English
- Prerequisites: Probability Theory (WTI and WTII) and basic knowledge of topology
Course description
Large deviations theory aims to characterise the exponential rate of convergence for sequences of probability measures. It has important applications to information theory and physics; in particular, large deviations theory explains the origin, meaning and role of entropy. Although a subbranch of probability theory, it can be rather analytic, using for example convexity arguments, topological arguments and some variational calculus. However, only WTI and WTII and basic knowledge of topology are required to follow the course. The course will cover both the fundamental theory as well as a number of important applications, like the local times of a random walk, large-time asymptotics of a Markov process and, if time allows, sequences of Markov processes.Format
Please contact me if you are interested in the course.Due to the corona outbreak, the course will be fully online. More precisely:
- Every Tuesday (starting 21st April) I will upload new lecture videos/pencasts and exercises on this website.
- The exercises are an essential part of the course. Please email me your solution of the exercises (or attempts thereof) by friday. A simple scan or photo of hand-written notes is fine, as long as it's readible!
- Every Tuesday 8:15--9:45 (starting 28th April) I will post a (Jitsi or Zoom) link, where anyone can join a question-and-answer session.
Exams
The exam will be held either in person (TU or WIAS), or online. I'm quite flexible (although I advice not to wait too long), but we'll need a second examinator, so it will be best to organise multiple exams on the same day. If you want to take the exam, please email me: 1) whether you want to do the exam in person or online, and 2) your preferred week.Weekly program
21st April | |
---|---|
|
König, Th. 1.4.3 den Hollander, Th. I.4 |
28st April | |
|
König, Th. 1.4.3 den Hollander, Th. I.4 |
5th May | |
|
den Hollander, Th. III.8 (Dembo & Zeitouni Lem. 4.1.4) den Hollander, Th. III.3 König, Lem. 2.1.5 & Bem. 2.1.6 (Dembo & Zeitouni Lem. 1.2.18) König, Lem. 2.1.3 (Dembo & Zeitouni Sect. 1.2) Dembo & Zeitouni, Lem. 4.1.23 |
12th May | |
|
König, Lem. 1.4.1.2(iii) & Satz 2.2.1 |
19th May | |
|
den Hollander, Th. II.1 & Lem. II.4 (combinatoric proof of Sanov on finite state space) König, Satz 2.4.1 & Lem. 2.4.3 (general proof of Sanov) (Dembo & Zeitouni Th. 6.2.10 (ridiculously general and advanced proof of Sanov)) |
25th May | |
|
König, Lem. 2.5.1 & Satz 2.5.2 den Hollander, Th. II.8 (Dembo & Zeitouni Th. 3.1.13 finite space & Cor. 6.5.10 on a Polish space) |
2nd June | |
|
König, Satz 3.3.1 & Satz 3.3.3 Dembo & Zeitouni Th. 4.3.1 & Th. 4.4.2 (den Hollander, Th. III.13) |
8th June | |
|
König, Lem. 3.4.1 & 3.4.3, Satz 3.4.4 Dembo & Zeitouni Th. 4.5.3 & Th. 4.5.20 (under relaxed assumptions) & Th. 4.5.27 den Hollander, Th. V.6 (in ℝd). |
16th June | |
|
König, Satz 3.5.7 & Satz 3.6.1 Dembo & Zeitouni Lem. 4.1.5(b) |
23rd June | |
|
Dembo & Zeitouni Cor. 4.2.6 & Th. 4.6.1 (See also Feng & Kurtz - Large deviations for stochastic processes, Th. 4.28 for a version in Skorohod space) |
30th June | |
|
Dembo & Zeitouni Section 5.1 & Th. 5.2.3 ( & Section 5.6) König Satz 2.3.1 & Satz 3.5.6 |
7th July | |
|
This is unfortunately not in the literature we've used so far Girsanov Theorem: Kipnis and Landim - Scaling Limits of Interacting Particle Systems (Prop. A.7.1) Empirical Process: Shwartz and Weiss - Large deviations for performance analysis (Th. 4.1) |
14th July | |
|