Course "Large Deviations", TU Berlin, Summer Semester 2020
- Lecturer: Michiel Renger
- Weekly hourse (SWS): 2.0
- 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 descriptionLarge 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.
FormatPlease 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.
ExamsThe 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.
|König, Th. 1.4.3
den Hollander, Th. I.4
|König, Th. 1.4.3
den Hollander, Th. I.4
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
|König, Lem. 18.104.22.168(iii) & Satz 2.2.1|
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))
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)
König, Satz 3.3.1 & Satz 3.3.3
Dembo & Zeitouni Th. 4.3.1 & Th. 4.4.2
(den Hollander, Th. III.13)
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).
König, Satz 3.5.7 & Satz 3.6.1
Dembo & Zeitouni Lem. 4.1.5(b)
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)
Dembo & Zeitouni Section 5.1 & Th. 5.2.3 ( & Section 5.6)
König Satz 2.3.1 & Satz 3.5.6
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)